From a6b0edaf6dd5f0e1c09b9ceeb387f84108f61fa4 Mon Sep 17 00:00:00 2001
From: scruel <scruelt@hotmail.com>
Date: Sun, 21 Jan 2018 23:48:08 +0800
Subject: [PATCH] optimize

fix and optimize the derivation of the cost function of week3
---
 image/20180120_105744.png | Bin 0 -> 73728 bytes
 image/20180121_110111.png | Bin 0 -> 76066 bytes
 index.html                |   3 ++-
 week2.html                |   4 ++--
 week2.md                  |   4 ++--
 week3.html                |  21 ++++++++++-----------
 week3.md                  |  23 +++++++++++------------
 week4.html                |   2 +-
 week4.md                  |   2 +-
 9 files changed, 29 insertions(+), 30 deletions(-)
 create mode 100644 image/20180120_105744.png
 create mode 100644 image/20180121_110111.png

diff --git a/image/20180120_105744.png b/image/20180120_105744.png
new file mode 100644
index 0000000000000000000000000000000000000000..097b24a104e41941f53a754fcc2368290945a727
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-<div  id='write'  class = 'is-node show-fences-line-number'><h1><a name='header-n284' class='md-header-anchor '></a>吴恩达(Andrew Ng)机器学习公开课中文笔记</h1><p><a href='https://github.com/scruel/ML-AndrewNg-Notes/'>GitHub 项目首页</a></p><p>&nbsp;</p><p><a href='./week1.html'>week1</a></p><ol start='' ><li>引言(Introduction)</li><li>单变量线性回归(Linear Regression with One Variable)</li></ol><p><a href='./week2.html'>week2</a></p><ol start='3' ><li>线性代数回顾(Linear Algebra Review)</li><li>多变量线性回归(Linear Regression with Multiple Variables)</li><li>Octave/Matlab 指南(Octave/Matlab Tutorial)</li></ol><p><a href='./week3.html'>week3</a></p><ol start='6' ><li>逻辑回归(Logistic Regression)</li><li>正则化(Regularization)</li></ol><p><a href='./week4.html'>week4</a></p><ol start='8' ><li>神经网络：表达(Neural Networks: Representation)</li></ol><p><a href='./week5.html'>week5</a></p><ol start='9' ><li>神经网络：学习(Neural Networks: Learning)</li></ol><p><a href='./week6.html'>week6</a></p><ol start='10' ><li>机器学习应用的建议(Advice for Applying Machine Learning)</li><li>机器学习系统设计(Machine Learning System Design)</li></ol><p><a href='./week7.html'>week7</a></p><ol start='12' ><li>支持向量机(Support Vector Machines)</li></ol><p><a href='./week8.html'>week8</a></p><ol start='13' ><li>无监督学习(Unsupervised Learning)</li><li>降维(Dimensionality Reduction)</li></ol><p><a href='./week9.html'>week9</a></p><ol start='15' ><li>异常检测(Anomaly Detection)</li><li>推荐系统(Recommender Systems)</li></ol><p><a href='./week10.html'>week10</a></p><ol start='17' ><li>大规模机器学习(Large Scale Machine Learning)</li></ol><p><a href='./week11.html'>week11</a></p><ol start='18' ><li>实战：图像光学识别(Application Example: Photo OCR)</li></ol><p>&nbsp;</p><p>&nbsp;</p><p><a href='http://creativecommons.org/licenses/by-nc/4.0/' target='_blank'><img src='https://i.creativecommons.org/l/by-nc/4.0/88x31.png' alt='Creative Commons License' /></a></p><p>This work is licensed under a <a href='http://creativecommons.org/licenses/by-nc/4.0/' target='_blank'>Creative Commons Attribution-NonCommercial 4.0 International License</a>.</p><p>&nbsp;</p><p><a href='https://zhuanlan.zhihu.com/p/32781741'>知乎文章</a></p><p>By: Scruel</p><p>&nbsp;</p><p><div style="display:none">
+<div  id='write'  class = 'is-node show-fences-line-number'><h1><a name='header-n136' class='md-header-anchor '></a>吴恩达(Andrew Ng)机器学习公开课中文笔记</h1><p><a href='https://github.com/scruel/ML-AndrewNg-Notes/'>GitHub 项目首页</a></p><p>&nbsp;</p><p><a href='./week1.html'>week1</a></p><ol start='' ><li>引言(Introduction)</li><li>单变量线性回归(Linear Regression with One Variable)</li></ol><p><a href='./week2.html'>week2</a></p><ol start='3' ><li>线性代数回顾(Linear Algebra Review)</li><li>多变量线性回归(Linear Regression with Multiple Variables)</li><li>Octave/Matlab 指南(Octave/Matlab Tutorial)</li></ol><p><a href='./week3.html'>week3</a></p><ol start='6' ><li>逻辑回归(Logistic Regression)</li><li>正则化(Regularization)</li></ol><p><a href='./week4.html'>week4</a></p><ol start='8' ><li>神经网络：表达(Neural Networks: Representation)</li></ol><p><a href='./week5.html'>week5</a></p><ol start='9' ><li>神经网络：学习(Neural Networks: Learning)</li></ol><p><a href='./week6.html'>week6</a></p><ol start='10' ><li>机器学习应用的建议(Advice for Applying Machine Learning)</li><li>机器学习系统设计(Machine Learning System Design)</li></ol><p><a href='./week7.html'>week7</a></p><ol start='12' ><li>支持向量机(Support Vector Machines)</li></ol><p><a href='./week8.html'>week8</a></p><ol start='13' ><li>无监督学习(Unsupervised Learning)</li><li>降维(Dimensionality Reduction)</li></ol><p><a href='./week9.html'>week9</a></p><ol start='15' ><li>异常检测(Anomaly Detection)</li><li>推荐系统(Recommender Systems)</li></ol><p><a href='./week10.html'>week10</a></p><ol start='17' ><li>大规模机器学习(Large Scale Machine Learning)</li></ol><p><a href='./week11.html'>week11</a></p><ol start='18' ><li>实战：图像光学识别(Application Example: Photo OCR)</li></ol><p>&nbsp;</p><p>&nbsp;</p><p><a href='http://creativecommons.org/licenses/by-nc/4.0/' target='_blank'><img src='https://i.creativecommons.org/l/by-nc/4.0/88x31.png' alt='Creative Commons License' /></a></p><p>This work is licensed under a <a href='http://creativecommons.org/licenses/by-nc/4.0/' target='_blank'>Creative Commons Attribution-NonCommercial 4.0 International License</a>.</p><p>&nbsp;</p><p><a href='https://zhuanlan.zhihu.com/p/32781741'>知乎文章</a></p><p>By: Scruel</p><p>&nbsp;</p><p><div style="display:none">
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-<div  id='write'  class = 'is-node show-fences-line-number'><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" href="#header-n2">4 多变量线性回归(Linear Regression with Multiple Variables)</a></span><span class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" href="#header-n3">4.1 多特征(Multiple Features)</a></span><span class="md-toc-item md-toc-h2" data-ref="n38"><a class="md-toc-inner" href="#header-n38">4.2 多变量梯度下降(Gradient Descent for Multiple Variables)</a></span><span class="md-toc-item md-toc-h2" data-ref="n63"><a class="md-toc-inner" href="#header-n63">4.3 梯度下降实践1-特征值缩放(Gradient Descent in Practice I - Feature Scaling)</a></span><span class="md-toc-item md-toc-h2" data-ref="n84"><a class="md-toc-inner" href="#header-n84">4.4 梯度下降实践2-学习速率(Gradient Descent in Practice II - Learning Rate)</a></span><span class="md-toc-item md-toc-h2" data-ref="n123"><a class="md-toc-inner" href="#header-n123">4.5 特征和多项式回归(Features and Polynomial Regression)</a></span><span class="md-toc-item md-toc-h2" data-ref="n138"><a class="md-toc-inner" href="#header-n138">4.6 正规方程(Normal Equation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n200"><a class="md-toc-inner" href="#header-n200">4.7 不可逆性正规方程(Normal Equation Noninvertibility)</a></span><span class="md-toc-item md-toc-h1" data-ref="n232"><a class="md-toc-inner" href="#header-n232">5 Octave/Matlab Tutorial</a></span><span class="md-toc-item md-toc-h2" data-ref="n235"><a class="md-toc-inner" href="#header-n235">5.1 Basic Operations</a></span><span class="md-toc-item md-toc-h2" data-ref="n236"><a class="md-toc-inner" href="#header-n236">5.2 Moving Data Around</a></span><span class="md-toc-item md-toc-h2" data-ref="n237"><a class="md-toc-inner" href="#header-n237">5.3 Computing on Data</a></span><span class="md-toc-item md-toc-h2" data-ref="n238"><a class="md-toc-inner" href="#header-n238">5.4 Plotting Data</a></span><span class="md-toc-item md-toc-h2" data-ref="n239"><a class="md-toc-inner" href="#header-n239">5.5 Control Statements: for, while, if statement</a></span><span class="md-toc-item md-toc-h2" data-ref="n240"><a class="md-toc-inner" href="#header-n240">5.6 向量化(Vectorization)</a></span><span class="md-toc-item md-toc-h2" data-ref="n243"><a class="md-toc-inner" href="#header-n243">5.x 常用函数整理</a></span></p></div><h1><a name='header-n2' class='md-header-anchor '></a>4 多变量线性回归(Linear Regression with Multiple Variables)</h1><h2><a name='header-n3' class='md-header-anchor '></a>4.1 多特征(Multiple Features)</h2><p>对于一个要度量的对象，一般来说会有不同维度的多个特征。比如之前的房屋价格预测例子中，除了房屋的面积大小，可能还有房屋的年限、房屋的层数等等其他特征：</p><p><img src='image/20180107_234509.png' alt='' /></p><p>这里由于特征不再只有一个，引入一些新的记号</p><blockquote><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-557-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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xlink:href="#E642-MJMATHI-6A" x="809" y="-430"></use></g></svg></span><script type="math/tex" id="MathJax-Element-521">{x}_{j}^{\left( i \right)}</script>: 代表样本矩阵中第 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-22-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="1.994ex" viewBox="0 -755.9 345.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E22-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 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600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-22">i</script> 个训练实例的第 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-57-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E643-MJMAIN-31" x="809" y="-435"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E643-MJMAIN-3D" x="8645" y="0"></use><g transform="translate(9701,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E643-MJMAIN-31"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E643-MJMAIN-34" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E643-MJMAIN-31" x="1001" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E643-MJMAIN-36" x="1501" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-522">{x}^{(2)}\text{=}\begin{bmatrix} 1416\\\ 3\\\ 2\\\ 40 \end{bmatrix}, {x}^{(2)}_{1} = 1416</script></p><p>多变量假设函数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-43-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.339ex" height="1.994ex" viewBox="0 -755.9 576.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E43-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMATHI-68" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-43">h</script> 表示为：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-523-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="37.425ex" height="2.577ex" viewBox="0 -806.1 16113.4 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E644-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" 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78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E644-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E644-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E644-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E644-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="1" id="E644-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E644-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E644-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="1" id="E644-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-3D" x="2804" y="0"></use><g transform="translate(3860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-30" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-2B" x="5006" y="0"></use><g transform="translate(6007,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-31" x="663" y="-213"></use></g><g transform="translate(6930,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-31" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-2B" x="8179" y="0"></use><g transform="translate(9179,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(10103,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-2B" x="11129" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-2E" x="11908" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-2E" x="12353" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-2E" x="12798" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMAIN-2B" x="13243" y="0"></use><g transform="translate(14022,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMATHI-6E" x="663" y="-213"></use></g><g transform="translate(15016,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E644-MJMATHI-6E" x="809" y="-213"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-523">h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}}+...+{\theta_{n}}{x_{n}}</script></p><p>对于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-44-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E44-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E44-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E44-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E44-MJMAIN-30" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-44">\theta_0</script>，和单特征中一样，我们将其看作基础数值。例如，房价的基础价格。</p><p>参数向量的维度为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-284-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.398ex" height="2.11ex" viewBox="0 -755.9 2323.9 908.7" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="1" id="E294-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E294-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="1" id="E294-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E294-MJMATHI-6E" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E294-MJMAIN-2B" x="822" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E294-MJMAIN-31" x="1823" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-284">n+1</script>，在特征向量中添加 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-524-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.384ex" height="1.76ex" viewBox="0 -504.6 1026.4 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E645-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E645-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E645-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E645-MJMAIN-30" x="809" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-524">x_{0}</script> 后，其维度也变为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-284-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.398ex" height="2.11ex" viewBox="0 -755.9 2323.9 908.7" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="1" id="E294-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E294-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="1" id="E294-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E294-MJMATHI-6E" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E294-MJMAIN-2B" x="822" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E294-MJMAIN-31" x="1823" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-284">n+1</script>， 则运用线性代数，可简化 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-43-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.339ex" height="1.994ex" viewBox="0 -755.9 576.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E43-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path></defs><g 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(h_\theta(x^{(i)}) - y^{(i)}) \cdot x_1^{(i)} \newline \; & \theta_2 := \theta_2 - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_2^{(i)} \newline & \vdots \newline \; & \theta_n := \theta_n - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_n^{(i)} &\newline \rbrace \end{aligned}</script></p><p>当然，同单变量梯度下降一样，计算时需要<strong>同时更新</strong>所有参数。</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-532-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.776ex" height="2.811ex" viewBox="0 -906.7 5500.9 1210.2" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E653-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 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id="MathJax-Element-535">x_0=1</script>）</p></blockquote><h2><a name='header-n63' class='md-header-anchor '></a>4.3 梯度下降实践1-特征值缩放(Gradient Descent in Practice I - Feature Scaling)</h2><p>在应用梯度下降算法实践时，由于各特征值的范围不一，可能会影响代价函数收敛速度。</p><p>以房价预测问题为例，这里选取房屋面积大小和房间数量这两个特征。</p><p>下图中，左图是以原始数据绘制的代价函数轮廓图，右图为采用特征缩放（都除以最大值）后图像。左图中呈现的图像较扁，相对于使用特征缩放方法的右图，梯度下降算法需要更多次的迭代。</p><p><img src='image/20180108_100751.png' alt='' /></p><p>&nbsp;</p><p>为了优化梯度下降的收敛速度，采用特征缩放的技巧，使各特征值的<strong>范围尽量一致</strong>。</p><p>除了以上图人工选择并除以一个参数的方式，<strong>均值归一化(Mean normalization)</strong>方法更为便捷，可采用它来对所有特征值统一缩放：</p><p> <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-536-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="31.999ex" height="4.445ex" viewBox="0 -1208.2 13777.2 1913.9" role="img" focusable="false" style="vertical-align: -1.639ex;"><defs><path stroke-width="1" id="E657-MJMATHI-78" d="M52 289Q59 331 106 386T222 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E657-MJMATHI-6D" x="6785" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E657-MJMATHI-69" x="7664" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E657-MJMATHI-6E" x="8009" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E657-MJMATHI-69" x="8610" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E657-MJMATHI-6D" x="8955" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E657-MJMATHI-75" x="9834" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E657-MJMATHI-6D" x="10406" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E657-MJMAIN-28" x="11285" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E657-MJMATHI-78" x="11674" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E657-MJMAIN-29" x="12247" y="0"></use></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E657-MJMAIN-2C" x="11546" y="0"></use><g transform="translate(11991,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">使</text></g><g transform="translate(12884,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">得</text></g></g></svg></span><script type="math/tex" id="MathJax-Element-536">x_i=\frac{x_i-average(x)}{maximum(x)-minimum(x)}, 使得 </script> <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-537-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.946ex" height="2.577ex" viewBox="0 -806.1 5143.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E658-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E658-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E658-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path stroke-width="1" id="E658-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E658-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E658-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E658-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="1" id="E658-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E658-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E658-MJMATHI-69" x="809" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E658-MJMAIN-2208" x="1194" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E658-MJMAIN-28" x="2139" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E658-MJMAIN-2212" x="2529" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E658-MJMAIN-31" x="3307" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E658-MJMAIN-2C" x="3808" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E658-MJMAIN-31" x="4253" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E658-MJMAIN-29" x="4754" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-537">x_i \in (-1,1)</script></p><p>对于特征的范围，并不一定需要使得 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-538-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.66ex" height="2.227ex" viewBox="0 -755.9 5020.1 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E659-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E659-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E659-MJAMS-2A7D" d="M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM94 170Q102 172 104 172Q110 171 254 103T535 -30T678 -98Q694 -106 694 -118Q694 -136 676 -138H670L382 -2Q92 135 90 137Q83 142 83 154Q84 164 94 170Z"></path><path stroke-width="1" id="E659-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E659-MJMAIN-2212" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E659-MJMAIN-31" x="778" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E659-MJAMS-2A7D" x="1556" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E659-MJMATHI-78" x="2613" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E659-MJAMS-2A7D" x="3463" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E659-MJMAIN-31" x="4519" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-538">-1 \leqslant x \leqslant 1</script>，类似于 <span 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xlink:href="#E660-MJMAIN-33" x="3741" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-539">1\leqslant x \leqslant 3</script> 等也是可取的，而诸如 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-540-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.309ex" height="2.227ex" viewBox="0 -755.9 7022.1 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E661-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E661-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E661-MJMAIN-30" d="M96 585Q152 666 249 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x="2781" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-541">-0.00001 \leqslant x \leqslant 0.00001</script>，就显得过大/过小了。</p><p>另外注意，一旦采用特征缩放，我们就需对所有的输入采用特征缩放，包括训练集、测试集、预测输入等。</p><h2><a name='header-n84' class='md-header-anchor '></a>4.4 梯度下降实践2-学习速率(Gradient Descent in Practice II - Learning Rate)</h2><p>通常，有两种方法来确定函数是否收敛</p><ul><li><p>多次迭代收敛法</p><ul><li>无法确定需要多少次迭代</li><li>较易绘制关于迭代次数的图像</li><li>根据图像易预测所需的迭代次数</li></ul></li><li><p>自动化测试收敛法（比较阈值）</p><ul><li>不易选取阈值</li><li>代价函数近乎直线时无法确定收敛情况</li></ul></li></ul><p>对于梯度下降，一般采用多次迭代收敛法来得出最小化代价函数的参数值，自动化测试收敛法（如设定 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-542-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.514ex" height="2.928ex" viewBox="0 -956.9 5388.1 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E663-MJMATHI-4A" d="M447 625Q447 637 354 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style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E72-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E72-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-72">\alpha</script>，一般上图展现的为适中情况，下图中，左图可能表明 <strong><span class="MathJax_Preview"></span><span class="MathJax_SVG" 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xlink:href="#E72-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-72">\alpha</script> 过大</strong>，代价函数<strong>无法收敛</strong>，右图可能表明 <strong><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-72-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E72-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E72-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-72">\alpha</script> 过小</strong>，代价函数<strong>收敛的太慢</strong>。当然，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-72-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E72-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E72-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-72">\alpha</script> 足够小时，代价函数在每轮迭代后一定会减少。</p><p><img src='image/20180108_104701.png' alt='' /></p><p>通过不断改变 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-72-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: 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style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E72-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E72-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-72">\alpha</script> 如 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-543-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="37.095ex" height="2.344ex" viewBox="0 -755.9 15971.3 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E664-MJMAIN-2026" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z"></path><path stroke-width="1" id="E664-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 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x="12073" y="0"></use><g transform="translate(12796,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E664-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E664-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E664-MJMAIN-31" x="779" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E664-MJMAIN-2C" x="14075" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E664-MJMAIN-2026" x="14798" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-543">\dots\;0,001,\;0.003,\;0.01,\;0.03,\;0.1,\;\dots</script></p><h2><a name='header-n123' class='md-header-anchor '></a>4.5 特征和多项式回归(Features and Polynomial Regression)</h2><p>在特征选取时，我们也可以自己归纳总结，定义一个新的特征，用来<strong>取代或拆分</strong>旧的一个或多个特征。比如，对于房屋面积特征来说，我们可以将其拆分为长度和宽度两个特征，反之，我们也可以合并长度和宽度这两个特征为面积这一个特征。</p><p>线性回归只能以直线来对数据进行拟合，有时候需要使用<strong>曲线</strong>来对数据进行拟合，即<strong>多项式回归(Polynomial Regression)</strong>。</p><p>比如一个二次方模型：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-544-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="25.85ex" height="3.044ex" viewBox="0 -906.7 11129.6 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E665-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" 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x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E665-MJMAIN-2B" x="8179" y="0"></use><g transform="translate(9179,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E665-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E665-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(10103,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E665-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E665-MJMAIN-32" x="809" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E665-MJMAIN-32" x="809" y="-435"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-544">h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}^2}</script></p><p>或者三次方模型：<span class="MathJax_Preview"></span><span class="MathJax_SVG" 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y="-213"></use></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E670-MJMATHI-4A" x="1370" y="0"></use><g transform="translate(2170,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E670-MJMAIN-28"></use><g transform="translate(389,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E670-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E670-MJMATHI-6A" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E670-MJMAIN-29" x="1250" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E670-MJMAIN-3D" x="4088" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E670-MJMAIN-30" x="5144" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-549">\frac{\partial}{\partial{\theta_{j}}}J\left( {\theta_{j}} \right)=0</script> ，通过解析函数的方式直接计算得出参数向量的值  <span 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y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-551">{X}^{-1}</script>: 矩阵 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-552-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.98ex" height="1.994ex" viewBox="0 -755.9 852.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E673-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E673-MJMATHI-58" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-552">X</script> 的逆，在 Octave 中，<code>inv</code> 函数用于计算矩阵的逆，类似的还有 <code>pinv</code> 函数。</p><p><code>X&#39;</code>: 在 Octave 中表示矩阵 X 的转置，即 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-553-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.386ex" height="2.344ex" viewBox="0 -906.7 1457.9 1009.2" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E674-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E674-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E674-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E674-MJMATHI-54" x="1215" y="513"></use></g></svg></span><script type="math/tex" id="MathJax-Element-553">X^T</script></p></blockquote><p>下表列出了正规方程法与梯度下降算法的对比</p><figure><table><thead><tr><th>条件</th><th>梯度下降</th><th>正规方程</th></tr></thead><tbody><tr><td>是否需要选取 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-72-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E72-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E72-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-72">\alpha</script></td><td>需要</td><td>不需要</td></tr><tr><td>是否需要迭代运算</td><td>需要</td><td>不需要</td></tr><tr><td>特征量大<sup><a href='#dfref-footnote-1' name='ref-footnote-1'>1</a></sup>时</td><td>适用，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-554-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.95ex" height="3.044ex" viewBox="0 -906.7 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288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="1" id="E675-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E675-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E675-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path 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137 -14T94 250Z"></path><path stroke-width="1" id="E676-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 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xlink:href="#E679-MJMATHI-54" x="550" y="583"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E679-MJMAIN-28" x="7492" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E679-MJMATHI-58" x="7882" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E679-MJMATHI-3B8" x="8734" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E679-MJMAIN-2212" x="9426" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E679-MJMATHI-79" x="10427" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E679-MJMAIN-29" x="10924" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-558">\begin{aligned} & J\left( \theta  \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {h_{\theta}}\left( {x^{(i)}} \right)-{y^{(i)}} \right)}^{2}}}\newline \; & =\frac{1}{2m}||X\theta-y||^2 \newline \; & =\frac{1}{2m}(X\theta-y)^T(X\theta-y) &\newline  \end{aligned}</script></p><p>展开上式可得</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-559-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="46.505ex" height="3.395ex" viewBox="0 -1007.2 20022.7 1461.5" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="1" id="E680-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E680-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 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x="2984" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E680-MJMATHI-3B8" x="3836" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E680-MJMAIN-2212" x="4528" y="0"></use><g transform="translate(5529,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E680-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E680-MJMATHI-54" x="663" y="612"></use></g><g transform="translate(6596,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E680-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E680-MJMATHI-54" x="1215" y="581"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E680-MJMATHI-79" x="8054" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E680-MJMAIN-2212" x="8774" y="0"></use><g transform="translate(9775,0)"><use 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y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-559">J(\theta )= \frac{1}{2m}\left( {{\theta }^{T}}{{X}^{T}}X\theta -{{\theta}^{T}}{{X}^{T}}y-{{y}^{T}}X\theta + {{y}^{T}}y \right)</script></p><p>注意到 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-560-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.021ex" height="3.044ex" viewBox="0 -1007.2 3023.1 1310.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E681-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E681-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E681-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E681-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 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x="1950" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-561">{{y}^{T}}X\theta</script> 都为标量，实际上是等价的，则</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-562-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="35.891ex" height="3.278ex" viewBox="0 -956.9 15452.8 1411.3" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="1" id="E683-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E683-MJMAIN-28" d="M94 250Q94 319 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMATHI-6D" x="500" y="0"></use></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMAIN-5B" x="4551" y="0"></use><g transform="translate(4829,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMATHI-54" x="1215" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMATHI-58" x="6287" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMATHI-3B8" x="7140" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMAIN-2212" x="7831" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMAIN-32" x="8832" y="0"></use><g transform="translate(9333,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMATHI-54" x="663" y="513"></use></g><g transform="translate(10400,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMATHI-54" x="1215" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMATHI-79" x="11858" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMAIN-2B" x="12578" y="0"></use><g transform="translate(13579,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMATHI-79" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMATHI-54" x="706" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMATHI-79" x="14676" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E683-MJMAIN-5D" x="15174" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-562">J(\theta) = \frac{1}{2m}[X^TX\theta-2\theta^TX^Ty+y^Ty]</script></p><p>接下来对<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-563-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.371ex" height="2.577ex" viewBox="0 -806.1 1882 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E684-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path 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725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E684-MJMATHI-4A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E684-MJMAIN-28" x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E684-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E684-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-563">J(\theta )</script> 求偏导，根据矩阵的求导法则:</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-564-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="21.356ex" height="3.745ex" viewBox="0 -1107.7 9195 1612.3" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path 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640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E685-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E685-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="1" id="E685-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 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630H244H224Q154 630 119 601Q101 585 93 554T81 486T76 443V437H36V443Z"></path><path stroke-width="1" id="E685-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="2654" height="60" x="0" y="220"></rect><g transform="translate(60,412)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMATHI-64" x="0" y="0"></use><g transform="translate(370,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMATHI-54" x="1215" y="581"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMATHI-41" x="1981" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMATHI-58" x="2731" y="0"></use></g><g transform="translate(840,-396)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMATHI-64" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMATHI-58" x="523" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMAIN-3D" x="3172" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMAIN-28" x="4228" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMATHI-41" x="4618" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMAIN-2B" x="5590" y="0"></use><g transform="translate(6591,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMAIN-54" x="1061" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMAIN-29" x="7952" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E685-MJMATHI-58" x="8342" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-564">\frac{dX^TAX}{dX}=(A+A^\mathrm{T})X</script></p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-565-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.164ex" height="3.745ex" viewBox="0 -1107.7 4376.3 1612.3" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E686-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="1" id="E686-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E686-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E686-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="1" id="E686-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="2051" height="60" x="0" y="220"></rect><g transform="translate(60,412)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E686-MJMATHI-64" x="0" y="0"></use><g transform="translate(370,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E686-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E686-MJMATHI-54" x="1215" y="581"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E686-MJMATHI-41" x="1981" y="0"></use></g><g transform="translate(539,-396)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E686-MJMATHI-64" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E686-MJMATHI-58" x="523" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E686-MJMAIN-3D" x="2569" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E686-MJMATHI-41" x="3625" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-565">\frac{dX^TA}{dX}={A}</script></p><p>所以有:</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-566-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="46.289ex" height="3.978ex" viewBox="0 -1208.2 19929.8 1712.8" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E687-MJMAIN-2202" d="M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 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xlink:href="#E689-MJMAIN-33"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E689-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E689-MJMAIN-32" x="779" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E689-MJMAIN-38" x="1279" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E689-MJMAIN-29" x="2169" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E689-MJMAIN-32" x="3618" y="675"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-568">{x_{1}}={x_{2}}*{{\left( 3.28 \right)}^{2}}</script>。</p></li><li><p>特征数量大于训练集的数量<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-569-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.343ex" height="2.577ex" viewBox="0 -806.1 3592.1 1109.7" role="img" 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48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-54" x="1215" y="513"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-58" x="1457" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-287">X^TX</script> 的结果不可逆，可尝试：</p><ul><li>减少多余/重复特征</li><li>增加训练集数量</li><li>使用正则化（后文）</li></ul><p>对于这类不可逆的矩阵，我们称之为<strong>奇异矩阵</strong>或<strong>退化矩阵</strong>。</p><p>这种情况下，如果还想使用正规方程法，在Octave中，可以选用 <code>pinv</code> 函数，<code>pinv</code> 区别于 <code>inv</code>，<code>pinv</code> 函数被称为伪逆函数，在矩阵不可逆的时候，使用这个函数仍可正确地计算出 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-70-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use 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-<div class='footnote-line'><span class='md-fn-count'>1</span> 一般来说，当 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-557-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.395ex" height="1.438ex" viewBox="0 -505.2 600.5 619.1" role="img" focusable="false" style="vertical-align: -0.264ex;"><defs><path stroke-width="1" id="E678-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E678-MJMATHI-6E" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-557">n</script> 超过 10000 时，对于正规方程而言，特征量较大。<a name='dfref-footnote-1' href='#ref-footnote-1' title='回到文档' class='reversefootnote' >↩</a></div>
+<div  id='write'  class = 'is-node show-fences-line-number'><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" href="#header-n2">4 多变量线性回归(Linear Regression with Multiple Variables)</a></span><span class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" href="#header-n3">4.1 多特征(Multiple Features)</a></span><span class="md-toc-item md-toc-h2" data-ref="n38"><a class="md-toc-inner" href="#header-n38">4.2 多变量梯度下降(Gradient Descent for Multiple Variables)</a></span><span class="md-toc-item md-toc-h2" data-ref="n63"><a class="md-toc-inner" href="#header-n63">4.3 梯度下降实践1-特征值缩放(Gradient Descent in Practice I - Feature Scaling)</a></span><span class="md-toc-item md-toc-h2" data-ref="n84"><a class="md-toc-inner" href="#header-n84">4.4 梯度下降实践2-学习速率(Gradient Descent in Practice II - Learning Rate)</a></span><span class="md-toc-item md-toc-h2" data-ref="n123"><a class="md-toc-inner" href="#header-n123">4.5 特征和多项式回归(Features and Polynomial Regression)</a></span><span class="md-toc-item md-toc-h2" data-ref="n138"><a class="md-toc-inner" href="#header-n138">4.6 正规方程(Normal Equation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n200"><a class="md-toc-inner" href="#header-n200">4.7 不可逆性正规方程(Normal Equation Noninvertibility)</a></span><span class="md-toc-item md-toc-h1" data-ref="n232"><a class="md-toc-inner" href="#header-n232">5 Octave/Matlab Tutorial</a></span><span class="md-toc-item md-toc-h2" data-ref="n235"><a class="md-toc-inner" href="#header-n235">5.1 Basic Operations</a></span><span class="md-toc-item md-toc-h2" data-ref="n236"><a class="md-toc-inner" href="#header-n236">5.2 Moving Data Around</a></span><span class="md-toc-item md-toc-h2" data-ref="n237"><a class="md-toc-inner" href="#header-n237">5.3 Computing on Data</a></span><span class="md-toc-item md-toc-h2" data-ref="n238"><a class="md-toc-inner" href="#header-n238">5.4 Plotting Data</a></span><span class="md-toc-item md-toc-h2" data-ref="n239"><a class="md-toc-inner" href="#header-n239">5.5 Control Statements: for, while, if statement</a></span><span class="md-toc-item md-toc-h2" data-ref="n240"><a class="md-toc-inner" href="#header-n240">5.6 向量化(Vectorization)</a></span><span class="md-toc-item md-toc-h2" data-ref="n243"><a class="md-toc-inner" href="#header-n243">5.x 常用函数整理</a></span></p></div><h1><a name='header-n2' class='md-header-anchor '></a>4 多变量线性回归(Linear Regression with Multiple Variables)</h1><h2><a name='header-n3' class='md-header-anchor '></a>4.1 多特征(Multiple Features)</h2><p>对于一个要度量的对象，一般来说会有不同维度的多个特征。比如之前的房屋价格预测例子中，除了房屋的面积大小，可能还有房屋的年限、房屋的层数等等其他特征：</p><p><img src='image/20180107_234509.png' alt='' /></p><p>这里由于特征不再只有一个，引入一些新的记号</p><blockquote><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-953-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-953">n</script>: 特征的总数 </p><p> <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-914-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.409ex" height="2.461ex" viewBox="0 -956.9 1467.6 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E1434-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 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id="MathJax-Element-914">{x}^{\left( i \right)}</script>: 代表样本矩阵中第 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-55-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="1.994ex" viewBox="0 -755.9 345.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E55-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g 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xlink:href="#E1435-MJMATHI-6A" x="809" y="-430"></use></g></svg></span><script type="math/tex" id="MathJax-Element-915">{x}_{j}^{\left( i \right)}</script>: 代表样本矩阵中第 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-55-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="1.994ex" viewBox="0 -755.9 345.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E55-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 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391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E57-MJMATHI-6A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-57">j</script> 列，也就是第 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-55-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="1.994ex" viewBox="0 -755.9 345.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E55-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E55-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-55">i</script> 个训练实例的第 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-57-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-11.5 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.027ex;"><defs><path stroke-width="1" id="E57-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E57-MJMATHI-6A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-57">j</script> 个特征。</p></blockquote><p>参照上图，则有 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-916-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="27.183ex" height="12.383ex" viewBox="0 -2917 11703.6 5331.5" role="img" focusable="false" style="vertical-align: -5.608ex;"><defs><path stroke-width="1" id="E1436-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 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y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-920">h</script> 表示为：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-918-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="37.425ex" height="2.577ex" viewBox="0 -806.1 16113.4 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E1438-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1438-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1438-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(10103,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1438-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1438-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1438-MJMAIN-2B" x="11129" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1438-MJMAIN-2E" x="11908" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1438-MJMAIN-2E" x="12353" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1438-MJMAIN-2E" x="12798" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1438-MJMAIN-2B" x="13243" y="0"></use><g transform="translate(14022,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1438-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1438-MJMATHI-6E" x="663" y="-213"></use></g><g transform="translate(15016,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1438-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1438-MJMATHI-6E" x="809" y="-213"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-918">h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}}+...+{\theta_{n}}{x_{n}}</script></p><p>对于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-182-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E184-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E184-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use 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554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E194-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="1" id="E194-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" 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151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E1439-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1439-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1448-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1448-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1448-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1448-MJMAIN-3D" x="2804" y="0"></use><g transform="translate(3860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1448-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1448-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1448-MJMATHI-78" x="4928" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-928">h_\theta\left(x\right)= \theta^T x</script>。</p><p>前文提到梯度下降对于最小化代价函数的通用性，则多变量梯度下降公式即</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" 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y="0"></use><g transform="translate(499,412)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1447-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1447-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1447-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1447-MJMAIN-29" x="14576" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1447-MJMAIN-22C5" x="15187" y="0"></use><g transform="translate(15688,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1447-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,521)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1447-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" 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Scaling)</h2><p>在应用梯度下降算法实践时，由于各特征值的范围不一，可能会影响代价函数收敛速度。</p><p>以房价预测问题为例，这里选取房屋面积大小和房间数量这两个特征。</p><p>下图中，左图是以原始数据绘制的代价函数轮廓图，右图为采用特征缩放（都除以最大值）后图像。左图中呈现的图像较扁，相对于使用特征缩放方法的右图，梯度下降算法需要更多次的迭代。</p><p><img src='image/20180108_100751.png' alt='' /></p><p>&nbsp;</p><p>为了优化梯度下降的收敛速度，采用特征缩放的技巧，使各特征值的<strong>范围尽量一致</strong>。</p><p>除了以上图人工选择并除以一个参数的方式，<strong>均值归一化(Mean normalization)</strong>方法更为便捷，可采用它来对所有特征值统一缩放：</p><p> <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-932-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="32.646ex" height="4.445ex" viewBox="0 -1208.2 14055.7 1913.9" role="img" focusable="false" style="vertical-align: -1.639ex;"><defs><path stroke-width="1" id="E1452-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 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transform="translate(397,0)"><rect stroke="none" width="9055" height="60" x="0" y="220"></rect><g transform="translate(2245,581)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-69" x="809" y="-213"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMAIN-2212" x="916" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-61" x="1695" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-76" x="2224" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-65" x="2710" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-72" x="3176" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-61" x="3628" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-67" x="4157" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-65" x="4638" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMAIN-28" x="5104" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-78" x="5494" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMAIN-29" x="6066" y="0"></use></g><g transform="translate(60,-436)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-61" x="878" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-78" x="1408" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-69" x="1980" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-6D" x="2326" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-75" x="3204" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-6D" x="3777" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMAIN-28" x="4655" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-78" x="5045" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMAIN-29" x="5617" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMAIN-2212" x="6007" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-6D" x="6785" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-69" x="7664" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-6E" x="8009" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-69" x="8610" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-6D" x="8955" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-75" x="9834" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-6D" x="10406" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMAIN-28" x="11285" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMATHI-78" x="11674" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMAIN-29" x="12247" y="0"></use></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1452-MJMAIN-2C" x="11824" y="0"></use><g transform="translate(12269,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">使</text></g><g transform="translate(13162,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">得</text></g></g></svg></span><script type="math/tex" id="MathJax-Element-932">x_i:=\frac{x_i-average(x)}{maximum(x)-minimum(x)}, 使得 </script> <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-933-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.946ex" height="2.577ex" viewBox="0 -806.1 5143.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E1453-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E1453-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E1453-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path stroke-width="1" id="E1453-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E1453-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E1453-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E1453-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="1" id="E1453-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1453-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1453-MJMATHI-69" x="809" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1453-MJMAIN-2208" x="1194" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1453-MJMAIN-28" x="2139" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1453-MJMAIN-2212" x="2529" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1453-MJMAIN-31" x="3307" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1453-MJMAIN-2C" x="3808" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1453-MJMAIN-31" x="4253" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1453-MJMAIN-29" x="4754" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-933">x_i \in (-1,1)</script></p><p>对于特征的范围，并不一定需要使得 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-934-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.66ex" height="2.227ex" viewBox="0 -755.9 5020.1 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E1454-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E1454-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E1454-MJAMS-2A7D" d="M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM94 170Q102 172 104 172Q110 171 254 103T535 -30T678 -98Q694 -106 694 -118Q694 -136 676 -138H670L382 -2Q92 135 90 137Q83 142 83 154Q84 164 94 170Z"></path><path stroke-width="1" id="E1454-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1454-MJMAIN-2212" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1454-MJMAIN-31" x="778" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1454-MJAMS-2A7D" x="1556" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1454-MJMATHI-78" x="2613" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1454-MJAMS-2A7D" x="3463" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1454-MJMAIN-31" x="4519" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-934">-1 \leqslant x \leqslant 1</script>，类似于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-935-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.852ex" height="2.227ex" viewBox="0 -755.9 4241.6 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E1455-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E1455-MJAMS-2A7D" d="M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM94 170Q102 172 104 172Q110 171 254 103T535 -30T678 -98Q694 -106 694 -118Q694 -136 676 -138H670L382 -2Q92 135 90 137Q83 142 83 154Q84 164 94 170Z"></path><path stroke-width="1" id="E1455-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E1455-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1455-MJMAIN-31" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1455-MJAMS-2A7D" x="778" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1455-MJMATHI-78" x="1834" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1455-MJAMS-2A7D" x="2684" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1455-MJMAIN-33" x="3741" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-935">1\leqslant x \leqslant 3</script> 等也是可取的，而诸如 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-936-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.309ex" height="2.227ex" viewBox="0 -755.9 7022.1 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E1456-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E1456-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 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id="E1456-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1456-MJMAIN-2212" x="0" y="0"></use><g transform="translate(778,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1456-MJMAIN-31"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1456-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1456-MJMAIN-30" x="1001" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1456-MJAMS-2A7D" x="2557" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1456-MJMATHI-78" x="3614" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1456-MJAMS-2A7D" x="4464" y="0"></use><g transform="translate(5520,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1456-MJMAIN-31"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1456-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1456-MJMAIN-30" x="1001" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-936">-100 \leqslant x \leqslant 100 </script>，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-937-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="24.578ex" height="2.227ex" viewBox="0 -755.9 10582.1 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E1457-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E1457-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E1457-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1457-MJMAIN-30" x="1279" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1457-MJMAIN-30" x="1780" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1457-MJMAIN-30" x="2280" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1457-MJMAIN-31" x="2781" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-937">-0.00001 \leqslant x \leqslant 0.00001</script>，就显得过大/过小了。</p><p>另外注意，一旦采用特征缩放，我们就需对所有的输入采用特征缩放，包括训练集、测试集、预测输入等。</p><h2><a name='header-n84' class='md-header-anchor '></a>4.4 梯度下降实践2-学习速率(Gradient Descent in Practice II - Learning Rate)</h2><p>通常，有两种方法来确定函数是否收敛</p><ul><li><p>多次迭代收敛法</p><ul><li>无法确定需要多少次迭代</li><li>较易绘制关于迭代次数的图像</li><li>根据图像易预测所需的迭代次数</li></ul></li><li><p>自动化测试收敛法（比较阈值）</p><ul><li>不易选取阈值</li><li>代价函数近乎直线时无法确定收敛情况</li></ul></li></ul><p>对于梯度下降，一般采用多次迭代收敛法来得出最小化代价函数的参数值，自动化测试收敛法（如设定 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-938-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.514ex" height="2.928ex" viewBox="0 -956.9 5388.1 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E1458-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E1458-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 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xlink:href="#E1458-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1458-MJMAIN-29" x="859" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1458-MJMAIN-3C" x="2326" y="0"></use><g transform="translate(3382,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1458-MJMAIN-31"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1458-MJMAIN-30" x="500" y="0"></use><g transform="translate(1001,393)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1458-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1458-MJMAIN-33" x="778" y="0"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-938">J\left(\theta\right) < {10}^{-3}</script> 时判定收敛）则几乎不会被使用。</p><p>我们可以通过绘制<strong>代价函数关于迭代次数的图像</strong>，可视化梯度下降的执行过程，借助直观的图形来发现代价函数趋向于多少时能趋于收敛，依据图像变化情况，确定诸如学习速率的取值，迭代次数的大小等问题。</p><p><img src='image/20180108_103357.png' alt='' /></p><p>对于学习速率 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-154-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E155-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 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26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E155-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-154">\alpha</script> 过大</strong>，代价函数<strong>无法收敛</strong>，右图可能表明 <strong><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-154-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E155-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E155-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-154">\alpha</script> 过小</strong>，代价函数<strong>收敛的太慢</strong>。当然，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-154-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E155-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E155-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-154">\alpha</script> 足够小时，代价函数在每轮迭代后一定会减少。</p><p><img src='image/20180108_104701.png' alt='' /></p><p>通过不断改变 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-154-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E155-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E155-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-154">\alpha</script> 值，绘制并观察图像，并以此来确定合适的学习速率。 尝试时可取 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-154-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E155-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E155-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-154">\alpha</script> 如 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-939-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="37.095ex" height="2.344ex" viewBox="0 -755.9 15971.3 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E1459-MJMAIN-2026" d="M78 60Q78 84 95 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xlink:href="#E1459-MJMAIN-2C" x="9570" y="0"></use><g transform="translate(10293,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1459-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1459-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1459-MJMAIN-30" x="779" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1459-MJMAIN-33" x="1279" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1459-MJMAIN-2C" x="12073" y="0"></use><g transform="translate(12796,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1459-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1459-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1459-MJMAIN-31" x="779" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1459-MJMAIN-2C" x="14075" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1459-MJMAIN-2026" x="14798" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-939">\dots\;0,001,\;0.003,\;0.01,\;0.03,\;0.1,\;\dots</script></p><h2><a name='header-n123' class='md-header-anchor '></a>4.5 特征和多项式回归(Features and Polynomial Regression)</h2><p>在特征选取时，我们也可以自己归纳总结，定义一个新的特征，用来<strong>取代或拆分</strong>旧的一个或多个特征。比如，对于房屋面积特征来说，我们可以将其拆分为长度和宽度两个特征，反之，我们也可以合并长度和宽度这两个特征为面积这一个特征。</p><p>线性回归只能以直线来对数据进行拟合，有时候需要使用<strong>曲线</strong>来对数据进行拟合，即<strong>多项式回归(Polynomial Regression)</strong>。</p><p>比如一个二次方模型：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-940-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="25.85ex" height="3.044ex" viewBox="0 -906.7 11129.6 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E1460-MJMATHI-68" d="M137 683Q138 683 209 688T282 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xlink:href="#E1460-MJMAIN-2B" x="5006" y="0"></use><g transform="translate(6007,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1460-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1460-MJMAIN-31" x="663" y="-213"></use></g><g transform="translate(6930,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1460-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1460-MJMAIN-31" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1460-MJMAIN-2B" x="8179" y="0"></use><g transform="translate(9179,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1460-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1460-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(10103,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1460-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1460-MJMAIN-32" x="809" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1460-MJMAIN-32" x="809" y="-435"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-940">h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}^2}</script></p><p>或者三次方模型：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-941-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="33.219ex" height="3.044ex" viewBox="0 -906.7 14302.4 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E1461-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-3D" x="2804" y="0"></use><g transform="translate(3860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-30" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-2B" x="5006" y="0"></use><g transform="translate(6007,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-31" x="663" y="-213"></use></g><g transform="translate(6930,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-31" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-2B" x="8179" y="0"></use><g transform="translate(9179,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(10103,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-32" x="809" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-32" x="809" y="-435"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-2B" x="11351" y="0"></use><g transform="translate(12352,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-33" x="663" y="-213"></use></g><g transform="translate(13275,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-33" x="809" y="510"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1461-MJMAIN-33" x="809" y="-434"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-941">h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}^2}+{\theta_{3}}{x_{3}^3}</script> </p><p>或者平方根模型： <span class="MathJax_Preview"></span><span class="MathJax_SVG" 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1462-MJMAIN-33" x="663" y="-213"></use></g><g transform="translate(13275,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1462-MJMAIN-221A" x="0" y="-140"></use><rect stroke="none" width="1026" height="60" x="833" y="601"></rect><g transform="translate(833,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1462-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1462-MJMAIN-33" x="809" y="-213"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-942">h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}^2}+{\theta_{3}}{\sqrt{x_3}}</script></p><p><img src='image/20180108_113132.png' alt='' /></p><p>在使用多项式回归时，要记住非常有必要进行特征缩放，比如 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-943-Frame" tabindex="-1" style="font-size: 100%; display: 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type="math/tex" id="MathJax-Element-944">x_1^2</script> 的范围则为 1- 1000000，不适用特征缩放的话，范围更有不一致，也更易影响效率。</p><h2><a name='header-n138' class='md-header-anchor '></a>4.6 正规方程(Normal Equation)</h2><p>对于一些线性回归问题来说，正规方程法给出了一个更好的解决问题的方式。</p><p>正规方程法，即令 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-945-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.111ex" height="3.978ex" viewBox="0 -1007.2 5645.1 1712.8" role="img" focusable="false" style="vertical-align: -1.639ex;"><defs><path stroke-width="1" id="E1465-MJMAIN-2202" d="M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 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617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1467-MJMATHI-58" x="0" y="0"></use><g transform="translate(859,410)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1467-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1467-MJMAIN-31" x="778" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-947">{X}^{-1}</script>: 矩阵 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-948-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.98ex" height="1.994ex" viewBox="0 -755.9 852.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E1468-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1468-MJMATHI-58" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-948">X</script> 的逆，在 Octave 中，<code>inv</code> 函数用于计算矩阵的逆，类似的还有 <code>pinv</code> 函数。</p><p><code>X&#39;</code>: 在 Octave 中表示矩阵 X 的转置，即 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-949-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.386ex" height="2.344ex" viewBox="0 -906.7 1457.9 1009.2" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E1469-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E1469-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 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id="MathJax-Element-949">X^T</script></p></blockquote><p>下表列出了正规方程法与梯度下降算法的对比</p><figure><table><thead><tr><th>条件</th><th>梯度下降</th><th>正规方程</th></tr></thead><tbody><tr><td>是否需要选取 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-154-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E155-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E155-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-154">\alpha</script></td><td>需要</td><td>不需要</td></tr><tr><td>是否需要迭代运算</td><td>需要</td><td>不需要</td></tr><tr><td>特征量大<sup><a href='#dfref-footnote-1' name='ref-footnote-1'>1</a></sup>时</td><td>适用，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-950-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.95ex" height="3.044ex" viewBox="0 -906.7 3423.1 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E1470-MJMATHI-4F" d="M740 435Q740 320 676 213T511 42T304 -22Q207 -22 138 35T51 201Q50 209 50 244Q50 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26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="1" id="E1470-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMAIN-32" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMATHI-6D" x="500" y="0"></use></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMAIN-28" x="3073" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMATHI-58" x="3462" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMATHI-3B8" x="4315" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMAIN-2212" x="5006" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMATHI-79" x="6007" y="0"></use><g transform="translate(6505,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMATHI-54" x="550" y="583"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMAIN-28" x="7492" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMATHI-58" x="7882" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMATHI-3B8" x="8734" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMAIN-2212" x="9426" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMATHI-79" x="10427" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1474-MJMAIN-29" x="10924" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-954">\begin{aligned} & J\left( \theta  \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {h_{\theta}}\left( {x^{(i)}} \right)-{y^{(i)}} \right)}^{2}}}\newline \; & =\frac{1}{2m}||X\theta-y||^2 \newline \; & =\frac{1}{2m}(X\theta-y)^T(X\theta-y) &\newline  \end{aligned}</script></p><p>展开上式可得</p><p><span 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xlink:href="#E1475-MJMATHI-79" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1475-MJMATHI-54" x="706" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1475-MJMATHI-58" x="10872" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1475-MJMATHI-3B8" x="11725" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1475-MJMAIN-2B" x="12417" y="0"></use><g transform="translate(13417,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1475-MJMATHI-79" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1475-MJMATHI-54" x="706" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1475-MJMATHI-79" x="14515" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1475-MJSZ1-29" x="15013" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-955">J(\theta )= \frac{1}{2m}\left( {{\theta }^{T}}{{X}^{T}}X\theta -{{\theta}^{T}}{{X}^{T}}y-{{y}^{T}}X\theta + {{y}^{T}}y \right)</script></p><p>注意到 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-956-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.021ex" height="3.044ex" viewBox="0 -1007.2 3023.1 1310.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E1476-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E1476-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E1476-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E1476-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1476-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1476-MJMATHI-54" x="663" y="612"></use><g transform="translate(1067,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1476-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1476-MJMATHI-54" x="1215" y="581"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1476-MJMATHI-79" x="2525" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-956">{{\theta}^{T}}{{X}^{T}}y</script> 与 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-957-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.62ex" height="2.811ex" viewBox="0 -906.7 2419.8 1210.2" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E1477-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E1477-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E1477-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E1477-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1477-MJMATHI-79" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1477-MJMATHI-54" x="706" y="513"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1477-MJMATHI-58" x="1097" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1477-MJMATHI-3B8" x="1950" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-957">{{y}^{T}}X\theta</script> 都为标量，实际上是等价的，则</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-958-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="35.891ex" height="3.278ex" viewBox="0 -956.9 15452.8 1411.3" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="1" id="E1478-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E1478-MJMAIN-28" d="M94 250Q94 319 104 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45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E1478-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="1" id="E1478-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-4A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMAIN-28" x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMAIN-29" x="1492" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMAIN-3D" x="2159" y="0"></use><g transform="translate(2938,0)"><g transform="translate(397,0)"><rect stroke="none" width="1095" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMAIN-31" x="524" y="572"></use><g transform="translate(60,-376)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMAIN-32" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-6D" x="500" y="0"></use></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMAIN-5B" x="4551" y="0"></use><g transform="translate(4829,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-54" x="1215" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-58" x="6287" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-3B8" x="7140" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMAIN-2212" x="7831" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMAIN-32" x="8832" y="0"></use><g transform="translate(9333,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-54" x="663" y="513"></use></g><g transform="translate(10400,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-54" x="1215" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-79" x="11858" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMAIN-2B" x="12578" y="0"></use><g transform="translate(13579,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-79" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-54" x="706" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMATHI-79" x="14676" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1478-MJMAIN-5D" x="15174" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-958">J(\theta) = \frac{1}{2m}[X^TX\theta-2\theta^TX^Ty+y^Ty]</script></p><p>接下来对<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-959-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.371ex" height="2.577ex" viewBox="0 -806.1 1882 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E1479-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E1479-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E1479-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E1479-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1479-MJMATHI-4A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1479-MJMAIN-28" x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1479-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1479-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-959">J(\theta )</script> 求偏导，根据矩阵的求导法则:</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-960-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="21.356ex" height="3.745ex" viewBox="0 -1107.7 9195 1612.3" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E1480-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="1" id="E1480-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E1480-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E1480-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="1" id="E1480-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E1480-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E1480-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="1" id="E1480-MJMAIN-54" d="M36 443Q37 448 46 558T55 671V677H666V671Q667 666 676 556T685 443V437H645V443Q645 445 642 478T631 544T610 593Q593 614 555 625Q534 630 478 630H451H443Q417 630 414 618Q413 616 413 339V63Q420 53 439 50T528 46H558V0H545L361 3Q186 1 177 0H164V46H194Q264 46 283 49T309 63V339V550Q309 620 304 625T271 630H244H224Q154 630 119 601Q101 585 93 554T81 486T76 443V437H36V443Z"></path><path stroke-width="1" id="E1480-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="2654" height="60" x="0" y="220"></rect><g transform="translate(60,412)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMATHI-64" x="0" y="0"></use><g transform="translate(370,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMATHI-54" x="1215" y="581"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMATHI-41" x="1981" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMATHI-58" x="2731" y="0"></use></g><g transform="translate(840,-396)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMATHI-64" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMATHI-58" x="523" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMAIN-3D" x="3172" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMAIN-28" x="4228" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMATHI-41" x="4618" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMAIN-2B" x="5590" y="0"></use><g transform="translate(6591,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMAIN-54" x="1061" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMAIN-29" x="7952" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1480-MJMATHI-58" x="8342" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-960">\frac{dX^TAX}{dX}=(A+A^\mathrm{T})X</script></p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-961-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.164ex" height="3.745ex" viewBox="0 -1107.7 4376.3 1612.3" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E1481-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="1" id="E1481-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E1481-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E1481-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="1" id="E1481-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="2051" height="60" x="0" y="220"></rect><g transform="translate(60,412)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1481-MJMATHI-64" x="0" y="0"></use><g transform="translate(370,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1481-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1481-MJMATHI-54" x="1215" y="581"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1481-MJMATHI-41" x="1981" y="0"></use></g><g transform="translate(539,-396)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1481-MJMATHI-64" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1481-MJMATHI-58" x="523" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1481-MJMAIN-3D" x="2569" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1481-MJMATHI-41" x="3625" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-961">\frac{dX^TA}{dX}={A}</script></p><p>所以有:</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-962-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="46.289ex" height="3.978ex" viewBox="0 -1208.2 19929.8 1712.8" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E1482-MJMAIN-2202" d="M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 372 401 400T333 428Q270 428 222 382Q197 354 183 323T150 221Q132 149 132 116Q132 21 232 21Q244 21 250 22Q327 35 374 112Q389 137 409 196T430 306Z"></path><path stroke-width="1" id="E1482-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E1482-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 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173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1432-MJMATHI-3B8" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1432-MJMAIN-3D" x="747" y="0"></use><g transform="translate(1803,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1432-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1432-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1432-MJMATHI-54" x="1215" y="583"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1432-MJMATHI-58" x="1916" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1432-MJSZ1-29" x="2768" y="-1"></use><g transform="translate(3227,619)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1432-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1432-MJMAIN-31" x="778" y="0"></use></g></g><g transform="translate(6035,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1432-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1432-MJMATHI-54" x="1215" y="583"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1432-MJMATHI-79" x="7493" y="0"></use></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-912">\theta ={{\left( {X^{T}}X \right)}^{-1}}{X^{T}}y</script></div><p>&nbsp;</p><h2><a name='header-n200' class='md-header-anchor '></a>4.7 不可逆性正规方程(Normal Equation Noninvertibility)</h2><p>（本部分内容为选讲）</p><p>正规方程无法应用于不可逆的矩阵，发生这种问题的概率很小，通常由于</p><ul><li><p>特征之间线性相关</p><p>比如同时包含英寸的尺寸和米为单位的尺寸两个特征，它们是线性相关的</p><p>即 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-964-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.059ex" height="3.044ex" viewBox="0 -1007.2 7344.7 1310.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E1484-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 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113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E1484-MJMAIN-2217" d="M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z"></path><path stroke-width="1" id="E1484-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1484-MJMAIN-33"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1484-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1484-MJMAIN-32" x="779" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1484-MJMAIN-38" x="1279" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1484-MJMAIN-29" x="2169" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1484-MJMAIN-32" x="3618" y="675"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-964">{x_{1}}={x_{2}}*{{\left( 3.28 \right)}^{2}}</script>。</p></li><li><p>特征数量大于训练集的数量<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-965-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.343ex" height="2.577ex" 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144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E1485-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1485-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1485-MJMATHI-6D" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1485-MJAMS-2A7D" x="1545" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1485-MJMATHI-6E" x="2602" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1485-MJMAIN-29" x="3202" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-965">\left(m \leqslant n \right)</script>。</p></li></ul><p>如果发现 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-195-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.366ex" height="2.344ex" viewBox="0 -906.7 2310.4 1009.2" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E197-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E197-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E197-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E197-MJMATHI-54" x="1215" y="513"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E197-MJMATHI-58" x="1457" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-195">X^TX</script> 的结果不可逆，可尝试：</p><ul><li>减少多余/重复特征</li><li>增加训练集数量</li><li>使用正则化（后文）</li></ul><p>对于这类不可逆的矩阵，我们称之为<strong>奇异矩阵</strong>或<strong>退化矩阵</strong>。</p><p>这种情况下，如果还想使用正规方程法，在Octave中，可以选用 <code>pinv</code> 函数，<code>pinv</code> 区别于 <code>inv</code>，<code>pinv</code> 函数被称为伪逆函数，在矩阵不可逆的时候，使用这个函数仍可正确地计算出 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-151-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E152-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-151">\theta</script> 的值。</p><h1><a name='header-n232' class='md-header-anchor '></a>5 Octave/Matlab Tutorial</h1><p>复习时可直接倍速回顾视频，笔记整理暂留。</p><h2><a name='header-n235' class='md-header-anchor '></a>5.1 Basic Operations</h2><h2><a name='header-n236' class='md-header-anchor '></a>5.2 Moving Data Around</h2><h2><a name='header-n237' class='md-header-anchor '></a>5.3 Computing on Data</h2><h2><a name='header-n238' class='md-header-anchor '></a>5.4 Plotting Data</h2><h2><a name='header-n239' class='md-header-anchor '></a>5.5 Control Statements: for, while, if statement</h2><h2><a name='header-n240' class='md-header-anchor '></a>5.6 向量化(Vectorization)</h2><p><span 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1486-MJMAIN-30" x="1191" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1486-MJMATHI-6E" x="545" y="1344"></use><g transform="translate(1362,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1486-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1486-MJMATHI-6A" x="663" y="-213"></use></g><g transform="translate(2223,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1486-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1486-MJMATHI-6A" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1486-MJMAIN-3D" x="3465" y="0"></use><g transform="translate(4522,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1486-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1486-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1486-MJMATHI-78" x="5589" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-966">\sum\limits_{j=0}^n\theta_jx_j=\theta^Tx</script></p><h2><a name='header-n243' class='md-header-anchor '></a>5.x 常用函数整理</h2><div class='footnotes-area'  ><hr/>
+<div class='footnote-line'><span class='md-fn-count'>1</span> 一般来说，当 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-953-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.395ex" height="1.438ex" viewBox="0 -505.2 600.5 619.1" role="img" focusable="false" style="vertical-align: -0.264ex;"><defs><path stroke-width="1" id="E1473-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1473-MJMATHI-6E" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-953">n</script> 超过 10000 时，对于正规方程而言，特征量较大。<a name='dfref-footnote-1' href='#ref-footnote-1' title='回到文档' class='reversefootnote' >↩</a></div>
 <div class='footnote-line'><span class='md-fn-count'>2</span> 梯度下降算法的普适性好，而对于特定的线性回归模型，正规方程是很好的替代品。<a name='dfref-footnote-2' href='#ref-footnote-2' title='回到文档' class='reversefootnote' >↩</a></div></div></div>
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diff --git a/week2.md b/week2.md
index 19df81c..7136e43 100644
--- a/week2.md
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@@ -79,9 +79,9 @@ $$
 
 除了以上图人工选择并除以一个参数的方式，**均值归一化(Mean normalization)**方法更为便捷，可采用它来对所有特征值统一缩放：
 
- $x_i=\frac{x_i-average(x)}{maximum(x)-minimum(x)}, 使得 $ $x_i \in (-1,1)$
+ $x_i:=\frac{x_i-average(x)}{maximum(x)-minimum(x)}, 使得 $ $x_i \in (-1,1)$
 
-对于特征的范围，并不一定需要使得 $-1 \leqslant x \leqslant 1$，类似于 $1\leqslant x \leqslant 3$ 等也是可取的，而诸如 $-100 \leqslant x \leqslant 100 $，$-0.00001 \leqslant x \leqslant 0.00001$，就显得过大/过小了。
+对于特征的范围，并不一定需要使得 $-1 \leqslant x \leqslant 1​$，类似于 $1\leqslant x \leqslant 3​$ 等也是可取的，而诸如 $-100 \leqslant x \leqslant 100 ​$，$-0.00001 \leqslant x \leqslant 0.00001​$，就显得过大/过小了。
 
 另外注意，一旦采用特征缩放，我们就需对所有的输入采用特征缩放，包括训练集、测试集、预测输入等。
 
diff --git a/week3.html b/week3.html
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@@ -332,19 +332,18 @@ header, .context-menu, .megamenu-content, footer { font-family: "Segoe UI", Aria
 </style>
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-<div  id='write'  class = 'is-node show-fences-line-number'><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n2">6 逻辑回归(Logistic Regression)</a></span><span class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n3">6.1 分类(Classification)</a></span><span class="md-toc-item md-toc-h2" data-ref="n40"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n40">6.2 假设函数表示(Hypothesis Representation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n64"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n64">6.3 决策边界(Decision Boundary)</a></span><span class="md-toc-item md-toc-h2" data-ref="n109"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n109">6.4 代价函数(Cost Function)</a></span><span class="md-toc-item md-toc-h2" data-ref="n138"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n138">6.5 简化的成本函数和梯度下降(Simplified Cost Function and Gradient Descent)</a></span><span class="md-toc-item md-toc-h2" data-ref="n201"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n201">6.6 进阶优化(Advanced Optimization)</a></span><span class="md-toc-item md-toc-h2" data-ref="n271"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n271">6.7 多类别分类: 一对多(Multiclass Classification: One-vs-all)</a></span><span class="md-toc-item md-toc-h1" data-ref="n289"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n289">7 正则化(Regularization)</a></span><span class="md-toc-item md-toc-h2" data-ref="n290"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n290">7.1 过拟合问题(The Problem of Overfitting)</a></span><span class="md-toc-item md-toc-h2" data-ref="n366"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n366">7.2 代价函数(Cost Function)</a></span><span class="md-toc-item md-toc-h2" data-ref="n423"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n423">7.3 线性回归正则化(Regularized Linear Regression)</a></span><span class="md-toc-item md-toc-h2" data-ref="n453"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n453">7.4 逻辑回归正则化(Regularized Logistic Regression)</a></span></p></div><h1><a name='header-n2' class='md-header-anchor '></a>6 逻辑回归(Logistic Regression)</h1><h2><a name='header-n3' class='md-header-anchor '></a>6.1 分类(Classification)</h2><p>在分类问题中，预测的结果是离散值（结果是否属于某一类），逻辑回归算法(Logistic Regression)被用于解决这类分类问题。</p><ul><li>垃圾邮件判断</li><li>金融欺诈判断</li><li>肿瘤诊断</li></ul><p>讨论肿瘤诊断问题：</p><p><img src='image/20180109_144040.png' alt='' /></p><p>肿瘤诊断问题的目的是告诉病人<strong>是否</strong>为恶性肿瘤，是一个<strong>二元分类问题(binary class problems)</strong>，则定义 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-172-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.68ex" height="2.577ex" 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xlink:href="#E181-MJMAIN-30" x="2221" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E181-MJMAIN-2C" x="2721" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E181-MJMAIN-31" x="3166" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E181-MJMAIN-7D" x="3667" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-172"> y \in\lbrace 0, 1\rbrace</script>，其中 0 表示<strong>负向类(negative class)</strong>，代表恶性肿瘤(&quot;-&quot;)，1 为<strong>正向类(positive class)</strong>，代表良性肿瘤(&quot;+&quot;)。如图，定义最右边的样本为<strong>偏差项</strong>。</p><p>在未加入偏差项时，线性回归算法给出了品红色的拟合直线，若规定</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-173-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.551ex" height="2.577ex" viewBox="0 -806.1 4973.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path 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xlink:href="#E183-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E183-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E183-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E183-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E183-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E183-MJAMS-2A7E" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E183-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E183-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E183-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-173">h_\theta(x) \geqslant 0.5</script> ，预测为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-221-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E231-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 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xlink:href="#E231-MJMAIN-31" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-221">y = 1</script>，即正向类；</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-175-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.551ex" height="2.577ex" viewBox="0 -806.1 4973.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E185-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 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xlink:href="#E185-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E185-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E185-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-175">h_\theta(x) \lt 0.5</script> ，预测为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-220-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E230-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E230-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E230-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E230-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E230-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E230-MJMAIN-30" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-220">y = 0</script>，即负向类。</p><p>即以 0.5 为<strong>阈值</strong>(threshold)，则我们就可以根据线性回归结果，得到相对正确的分类结果 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-27-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.155ex" height="1.877ex" viewBox="0 -504.6 497.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E27-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E27-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-27">y</script>。</p><p>&nbsp;</p><p>接下来加入偏差项，线性回归算法给出了靛青色的拟合直线，如果阈值仍然为 0.5，可以看到算法在某些情况下会给出完全错误的结果，对于癌症、肿瘤诊断这类要求预测极其精确的问题，这种情况是无法容忍的。</p><p>不仅如此，线性回归算法的值域为全体实数集（<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-177-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.086ex" height="2.577ex" viewBox="0 -806.1 4342.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E187-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E187-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E187-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E187-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 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y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-177">h_\theta(x) \in R</script>），则当线性回归函数给出诸如 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-178-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="31.626ex" height="2.577ex" viewBox="0 -806.1 13616.8 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E188-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 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xlink:href="#E189-MJMAIN-7B" x="1720" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E189-MJMAIN-30" x="2221" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E189-MJMAIN-2C" x="2721" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E189-MJMAIN-31" x="3166" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E189-MJMAIN-7D" x="3667" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-179">y \in \lbrace 0, 1\rbrace</script>，这显得非常怪异。</p><p>&nbsp;</p><p>区别于线性回归算法，逻辑回归算法是一个分类算法，<strong>其输出值永远在 0 到 1 之间</strong>，即 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-180-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.49ex" height="2.577ex" viewBox="0 -806.1 5808.2 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E190-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E190-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 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284 52 289Z"></path><path stroke-width="1" id="E190-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E190-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path stroke-width="1" id="E190-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 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x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E190-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E190-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E190-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E190-MJMAIN-2208" x="2637" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E190-MJMAIN-28" x="3583" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E190-MJMAIN-30" x="3972" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E190-MJMAIN-2C" x="4473" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E190-MJMAIN-31" x="4918" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E190-MJMAIN-29" x="5418" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-180">h_\theta(x) \in (0,1)</script>。</p><h2><a name='header-n40' class='md-header-anchor '></a>6.2 假设函数表示(Hypothesis Representation)</h2><p>为了使 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-181-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.49ex" height="2.577ex" viewBox="0 -806.1 5808.2 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E191-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E1933-MJSZ1-29" x="2098" y="-1"></use></g></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-1300">h_\theta \left( x \right)=g(z)=g\left(\theta^{T}x \right)</script></div><p>对比线性回归函数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-182-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.776ex" height="2.811ex" viewBox="0 -906.7 5500.9 1210.2" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E192-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 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y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E192-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E192-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E192-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E192-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E192-MJMAIN-3D" x="2804" y="0"></use><g transform="translate(3860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E192-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E192-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E192-MJMATHI-78" x="4928" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-182">h_\theta \left( x \right)=\theta^{T}x</script>，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-92-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.116ex" height="1.877ex" viewBox="0 -504.6 480.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E92-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E92-MJMATHI-67" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-92">g</script> 表示逻辑函数(<a href='https://en.wikipedia.org/wiki/Logistic_function' target='_blank'>logistic function</a>)，复合起来，则称为逻辑回归函数。</p><p>逻辑函数是 S 形函数，会将所有实数映射到 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-183-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.168ex" height="2.577ex" viewBox="0 -806.1 2225.2 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E193-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 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1)</script> 范围。</p><p><a href='https://en.wikipedia.org/wiki/Sigmoid_function' target='_blank'>sigmoid 函数</a>（如下图）是逻辑函数的特殊情况，其公式为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-184-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.814ex" height="3.395ex" viewBox="0 -956.9 5517.2 1461.5" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E194-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 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y="0"></use><g transform="translate(329,204)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E194-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E194-MJMATHI-7A" x="778" y="0"></use></g></g></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-184">g\left( z \right)=\frac{1}{1+{{e}^{-z}}}</script>。 </p><p><img src='image/2413fbec8ff9fa1f19aaf78265b8a33b_Logistic_function.png' alt='sigmoid function' /></p><p>应用 sigmoid 函数，则逻辑回归模型：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-185-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="25.328ex" height="3.978ex" viewBox="0 -956.9 10905.2 1712.8" role="img" focusable="false" style="vertical-align: -1.756ex;"><defs><path stroke-width="1" id="E195-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 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xlink:href="#E195-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMAIN-3D" x="2637" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMATHI-67" x="3694" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMAIN-28" x="4174" y="0"></use><g transform="translate(4564,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMATHI-78" x="5631" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMAIN-29" x="6204" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMAIN-3D" x="6871" y="0"></use><g transform="translate(7649,0)"><g transform="translate(397,0)"><rect stroke="none" width="2737" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMAIN-31" x="1685" y="572"></use><g transform="translate(60,-608)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMAIN-2B" x="500" y="0"></use><g transform="translate(904,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMATHI-65" x="0" y="0"></use><g transform="translate(329,204)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMAIN-2212" x="0" y="0"></use><g transform="translate(389,0)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMATHI-54" x="469" y="319"></use></g><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMATHI-78" x="2052" y="0"></use></g></g></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-185">h_{\theta}(x)=g(\theta^Tx) =\frac{1}{1+e^{-\theta^Tx}}</script></p><p>逻辑回归模型中，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-186-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.868ex" height="2.577ex" viewBox="0 -806.1 2526.7 1109.7" 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103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E12-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-12">x</script> 以及参数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-70-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-70">\theta</script>，计算得出”输出 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-217-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E227-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E227-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E227-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMAIN-31" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-217">y=1</script>“的可能性(estimated probability)，概率学中表示为：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-188-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="42.566ex" height="5.846ex" 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664T200 712Q193 712 186 709T167 698T147 668T134 615Q132 595 132 548V527Q132 436 165 403Q183 385 203 385H207ZM500 146Q500 234 544 290T647 347Q699 347 737 292T776 146T737 0T646 -56Q590 -56 545 0T500 146ZM651 -18Q679 -18 707 24T736 146Q736 215 711 262T644 309Q637 309 630 306T611 295T591 265T578 212Q577 200 577 146V124Q577 -18 647 -18H651Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-37"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-25" x="1001" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-190">70\%</script> 的概率得了恶性肿瘤。</p><h2><a name='header-n64' class='md-header-anchor '></a>6.3 决策边界(Decision Boundary)</h2><p>决策边界的概念，可帮助我们更好地理解逻辑回归模型的拟合原理。</p><p>在逻辑回归中，有假设函数 <span 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xlink:href="#E201-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E201-MJMATHI-78" x="1526" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E201-MJSZ1-29" x="2098" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-191">h_\theta \left( x \right)=g(z)=g\left(\theta^{T}x \right)</script>。</p><p>为了得出分类的结果，这里和前面一样，规定以 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-192-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="1.994ex" viewBox="0 -755.9 1279.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E202-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E203-MJMAIN-2192" x="5251" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E203-MJMATHI-79" x="6529" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E203-MJMAIN-3D" x="7304" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E203-MJMAIN-30" x="8361" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-193">\begin{align*}& h_\theta(x) \geq 0.5 \rightarrow y = 1 \newline& h_\theta(x) < 0.5 \rightarrow y = 0 \newline\end{align*}</script></p><p>回忆一下 sigmoid 函数的图像：</p><p><img src='image/2413fbec8ff9fa1f19aaf78265b8a33b_Logistic_function.png' alt='sigmoid function' /></p><p>观察可得当 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-194-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.084ex" height="2.577ex" viewBox="0 -806.1 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-14T94 250Z"></path><path stroke-width="1" id="E204-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E204-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" 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210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E204-MJMATHI-67" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E204-MJMAIN-28" x="480" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E204-MJMATHI-7A" x="870" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E204-MJMAIN-29" x="1338" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E204-MJMAIN-2265" x="2005" y="0"></use><g transform="translate(3062,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E204-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E204-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E204-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-194">g(z) \geq 0.5</script> 时，有 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-203-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.349ex" height="2.227ex" viewBox="0 -755.9 2303.1 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E213-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E213-MJMAIN-2265" d="M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z"></path><path stroke-width="1" id="E213-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 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style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E206-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E206-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 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xlink:href="#E206-MJMATHI-78" x="1067" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E206-MJMAIN-2265" x="1917" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E206-MJMAIN-30" x="2974" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-196">\theta^Tx \geq 0</script>。</p><p>同线性回归模型的不同点在于： <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-197-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="31.512ex" height="5.846ex" viewBox="0 -1509.8 13567.4 2517" role="img" focusable="false" style="vertical-align: -2.339ex;"><defs><path stroke-width="1" id="E207-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 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xlink:href="#E207-MJMAIN-31" x="12747" y="0"></use></g><g transform="translate(50,-700)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMATHI-7A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-2192" x="746" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-2212" x="2024" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-221E" x="2803" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-2C" x="3803" y="0"></use><g transform="translate(4248,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMATHI-65" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-221E" x="659" y="583"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-2192" x="5800" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-221E" x="7078" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-21D2" x="8357" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMATHI-67" x="9635" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-28" x="10115" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMATHI-7A" x="10505" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-29" x="10973" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-3D" x="11641" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-30" x="12697" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-197">\begin{align*}z \to +\infty, e^{-\infty} \to 0 \Rightarrow g(z)=1 \newline z \to -\infty, e^{\infty}\to \infty \Rightarrow g(z)=0 \end{align*}</script></p><p>直观一点来个例子，<span 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{\theta_0}+{\theta_1}{x_1}+{\theta_{2}}{x_{2}}\geq0</script> 时，就预测 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-221-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E231-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 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x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E231-MJMAIN-31" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-221">y = 1</script>，即预测为正向类。</p><p>如果取 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-201-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.001ex" height="9.114ex" viewBox="0 -2213.4 4736.6 3924.2" role="img" focusable="false" style="vertical-align: -3.974ex;"><defs><path stroke-width="1" id="E211-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E211-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E211-MJMAIN-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path stroke-width="1" id="E211-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E211-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="1" id="E211-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E211-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path stroke-width="1" id="E211-MJSZ4-23A1" d="M319 -645V1154H666V1070H403V-645H319Z"></path><path stroke-width="1" id="E211-MJSZ4-23A3" d="M319 -644V1155H403V-560H666V-644H319Z"></path><path stroke-width="1" id="E211-MJSZ4-23A2" d="M319 0V602H403V0H319Z"></path><path stroke-width="1" id="E211-MJSZ4-23A4" d="M0 1070V1154H347V-645H263V1070H0Z"></path><path stroke-width="1" id="E211-MJSZ4-23A6" d="M263 -560V1155H347V-644H0V-560H263Z"></path><path stroke-width="1" id="E211-MJSZ4-23A5" d="M263 0V602H347V0H263Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E211-MJMATHI-3B8" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E211-MJMAIN-3D" x="747" y="0"></use><g transform="translate(1803,0)"><g transform="translate(0,2150)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E211-MJSZ4-23A1" x="0" y="-1155"></use><g transform="translate(0,-2048.5066225165565) scale(1,0.49337748344370863)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E211-MJSZ4-23A2"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E211-MJSZ4-23A3" x="0" y="-3155"></use></g><g transform="translate(834,0)"><g transform="translate(-15,0)"><g transform="translate(0,1350)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E211-MJMAIN-2212" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E211-MJMAIN-33" x="778" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E211-MJMAIN-31" x="389" y="-50"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E211-MJMAIN-31" x="389" y="-1450"></use></g></g><g transform="translate(2265,2150)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E211-MJSZ4-23A4" x="0" y="-1155"></use><g transform="translate(0,-2048.5066225165565) scale(1,0.49337748344370863)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E211-MJSZ4-23A5"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E211-MJSZ4-23A6" x="0" y="-3155"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-201">\theta = \begin{bmatrix} -3\\1\\1\end{bmatrix}</script>，则有 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-202-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.606ex" height="2.227ex" viewBox="0 -755.9 7580.3 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E212-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E212-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E212-MJMAIN-33" x="2581" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E212-MJMAIN-2B" x="3303" y="0"></use><g transform="translate(4304,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E212-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E212-MJMAIN-31" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E212-MJMAIN-2B" x="5553" y="0"></use><g transform="translate(6553,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E212-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E212-MJMAIN-32" x="809" y="-213"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-202">z = -3+{x_1}+{x_2}</script>，当 <span class="MathJax_Preview"></span><span class="MathJax_SVG" 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70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E214-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E214-MJMAIN-31" x="809" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E214-MJMAIN-2B" x="1248" y="0"></use><g transform="translate(2249,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E214-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E214-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E214-MJMAIN-2265" x="3553" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E214-MJMAIN-33" x="4609" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-204">{x_1}+{x_2} \geq 3</script> 时，易绘制图中的品红色直线即<strong>决策边界</strong>，为正向类（以红叉标注的数据）给出 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-217-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E227-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E227-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E227-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 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175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="1" id="E216-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path stroke-width="1" id="E216-MJSZ1-28" d="M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 56T152 251Z"></path><path stroke-width="1" id="E216-MJSZ1-29" d="M305 251Q305 -145 69 -349H56Q43 -349 39 -347T35 -338Q37 -333 60 -307T108 -239T160 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMAIN-2B" x="4776" y="0"></use><g transform="translate(5777,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(6701,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMAIN-2B" x="7949" y="0"></use><g transform="translate(8950,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMAIN-33" x="663" y="-213"></use></g><g transform="translate(9873,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMAIN-32" x="809" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMAIN-31" x="809" y="-435"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMAIN-2B" x="11122" y="0"></use><g transform="translate(12123,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMAIN-34" x="663" y="-213"></use></g><g transform="translate(13046,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMAIN-32" x="809" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJMAIN-32" x="809" y="-435"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E216-MJSZ1-29" x="14072" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-206">{h_\theta}\left( x \right)=g\left( {\theta_0}+{\theta_1}{x_1}+{\theta_{2}}{x_{2}}+{\theta_{3}}x_{1}^{2}+{\theta_{4}}x_{2}^{2} \right)</script></p><p>这里取 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-207-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.001ex" height="15.651ex" viewBox="0 -3620.7 4736.6 6738.7" role="img" focusable="false" style="vertical-align: -7.242ex;"><defs><path stroke-width="1" id="E217-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E217-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E217-MJMAIN-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path stroke-width="1" id="E217-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E217-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path 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xlink:href="#E217-MJMAIN-31" x="778" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E217-MJMAIN-30" x="389" y="1350"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E217-MJMAIN-30" x="389" y="-50"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E217-MJMAIN-31" x="389" y="-1450"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E217-MJMAIN-31" x="389" y="-2850"></use></g></g><g transform="translate(2265,3550)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E217-MJSZ4-23A4" x="0" y="-1155"></use><g transform="translate(0,-4843.8708609271525) scale(1,5.129139072847682)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E217-MJSZ4-23A5"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E217-MJSZ4-23A6" x="0" y="-5955"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-207">\theta = \begin{bmatrix} -1\\0\\0\\1\\1\end{bmatrix}</script>，决策边界对应了一个在原点处的单位圆（<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-208-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.978ex" height="2.577ex" viewBox="0 -906.7 6018.1 1109.7" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E218-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 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xlink:href="#E218-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E218-MJMAIN-32" x="809" y="-213"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E218-MJMAIN-32" x="1451" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E218-MJMAIN-3D" x="4461" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E218-MJMAIN-31" x="5517" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-208">{x_1}^2+{x_2}^2 = 1</script>），如此便可给出分类结果，如图中品红色曲线：</p><p>&nbsp;</p><p><img src='image/20180111_000653.png' alt='' /></p><p>当然，通过一些更为复杂的多项式，还能拟合那些图像显得非常怪异的数据，使得决策边界形似碗状、爱心状等等。</p><p>&nbsp;</p><p>简单来说，决策边界就是<strong>分类的分界线</strong>，分类现在实际就由 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-209-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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xlink:href="#E219-MJMATHI-7A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-209">z</script> (中的 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-70-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 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36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-70">\theta</script> 多少时能很好的拟合数据？当然，见招拆招，总要来个 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-249-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.371ex" height="2.577ex" viewBox="0 -806.1 1882 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E259-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 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width="4.371ex" height="2.577ex" viewBox="0 -806.1 1882 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E259-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E259-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E259-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 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xlink:href="#E259-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E259-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-249">J(\theta)</script> 的图像，如下图</p><p><img src='image/20180111_080314.png' alt='' /></p><p>回忆线性回归中的平方损失函数，其是一个二次凸函数（碗状），二次凸函数的重要性质是只有一个局部最小点即全局最小点。上图中有许多局部最小点，这样将使得梯度下降算法无法确定收敛点是全局最优。</p><p><img src='image/20180111_080514.png' alt='' /></p><p>如果此处的损失函数也是一个凸函数，是否也有同样的性质，从而最优化？这类讨论凸函数最优值的问题，被称为<strong>凸优化问题(Convex optimization)</strong>。</p><p>当然，损失函数不止平方损失函数一种。</p><p>对于逻辑回归，更换平方损失函数为<strong>对数损失函数</strong>，可由统计学中的最大似然估计方法推出代价函数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-249-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.371ex" height="2.577ex" viewBox="0 -806.1 1882 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-28" x="2007" y="0"></use><g transform="translate(2396,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMATHI-3B8" x="815" y="-219"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-28" x="3404" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMATHI-78" x="3794" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-29" x="4366" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-2C" x="4756" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMATHI-79" x="5201" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-29" x="5699" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-3D" x="6366" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-2212" x="7422" y="0"></use><g transform="translate(8367,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-6C"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-6F" x="278" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-67" x="779" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-28" x="9647" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-31" x="10036" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-2212" x="10759" y="0"></use><g transform="translate(11760,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMATHI-3B8" x="815" y="-219"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-28" x="12768" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMATHI-78" x="13158" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-29" x="13730" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-29" x="14120" y="0"></use></g></g><g transform="translate(16772,0)"><g transform="translate(0,-881)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-69"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-66" x="278" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-79" x="835" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-3D" x="1613" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-31" x="2642" y="0"></use></g><g transform="translate(0,-2232)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-69"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-66" x="278" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-79" x="835" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-3D" x="1613" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E225-MJMAIN-30" x="2642" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-215">\begin{align*}& J(\theta) = \dfrac{1}{m} \sum_{i=1}^m \mathrm{Cost}(h_\theta(x^{(i)}),y^{(i)}) \newline & \mathrm{Cost}(h_\theta(x),y) = -\log(h_\theta(x)) \; & \text{if y = 1} \newline & \mathrm{Cost}(h_\theta(x),y) = -\log(1-h_\theta(x)) \; & \text{if y = 0}\end{align*}</script></p><p>则有关于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-249-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.371ex" height="2.577ex" viewBox="0 -806.1 1882 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E259-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E259-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E259-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E259-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E259-MJMATHI-4A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E259-MJMAIN-28" x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E259-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E259-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-249">J(\theta)</script> 的图像如下：</p><p><img src='image/20180111_080614.png' alt='' /></p><p>如左图，当训练集的结果为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-217-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E227-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E227-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E227-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMAIN-31" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-217">y=1</script>（正样本）时，随着假设函数趋向于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-109-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.162ex" height="1.994ex" viewBox="0 -755.9 500.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E109-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E109-MJMAIN-31" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-109">1</script>，代价函数的值会趋于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-45-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.162ex" height="1.994ex" viewBox="0 -755.9 500.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E45-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E45-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-45">0</script>，即意味着拟合程度很好。如果假设函数此时趋于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-45-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.162ex" height="1.994ex" viewBox="0 -755.9 500.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E45-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E45-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-45">0</script>，则会给出一个<strong>很高的代价</strong>，拟合程度<strong>差</strong>，算法会根据其迅速纠正 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-70-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-70">\theta</script> 值，右图 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-218-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E228-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E228-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E228-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E228-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E228-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E228-MJMAIN-30" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-218">y=0</script> 同理。</p><p>区别于平方损失函数，对数损失函数也是一个凸函数，但没有局部最优值。</p><h2><a name='header-n138' class='md-header-anchor '></a>6.5 简化的成本函数和梯度下降(Simplified Cost Function and Gradient Descent)</h2><p>由于懒得分类讨论，对于二元分类问题，我们可把代价函数<strong>简化</strong>为一个函数： 
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xlink:href="#E229-MJMAIN-29" x="2916" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E229-MJMAIN-2212" x="15450" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E229-MJMAIN-28" x="16451" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E229-MJMAIN-31" x="16840" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E229-MJMAIN-2212" x="17563" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E229-MJMATHI-79" x="18564" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E229-MJMAIN-29" x="19061" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E229-MJMAIN-D7" x="19673" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E229-MJMATHI-6C" x="20674" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E229-MJMATHI-6F" x="20972" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E229-MJMAIN-29" x="4639" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-219">Cost\left( {h_\theta}\left( x \right),y \right)=-y\times log\left( {h_\theta}\left( x \right) \right)-(1-y)\times log\left( 1-{h_\theta}\left( x \right) \right)</script></p><p>当 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-220-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E230-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 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16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E230-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E230-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E230-MJMAIN-30" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-220">y = 0</script>，左边式子整体为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-45-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.162ex" height="1.994ex" viewBox="0 -755.9 500.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E45-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E45-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-45">0</script>，当 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-221-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E231-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 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y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E238-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E238-MJMATHI-78" x="1526" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E238-MJSZ1-29" x="2098" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-228">h_\theta(x) = g\left(\theta^{T}x \right)</script>，不过求导后的结果也相同。</p><p>向量化实现：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-229-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="26.233ex" height="3.161ex" viewBox="0 -906.7 11294.5 1361" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="1" id="E239-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 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390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path><path stroke-width="1" id="E239-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 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1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E239-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" 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y="-488"></use></g></g><g transform="translate(4755,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E239-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E239-MJMATHI-54" x="1215" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E239-MJMAIN-28" x="6213" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E239-MJMATHI-67" x="6603" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E239-MJMAIN-28" x="7083" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E239-MJMATHI-58" x="7473" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E239-MJMATHI-3B8" x="8325" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E239-MJMAIN-29" x="8795" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E239-MJMAIN-2212" x="9406" y="0"></use><use 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127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E240-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E240-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E240-MJMAIN-29" d="M60 749L64 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E259-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E259-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-249">J(\theta)</script> 值。</p><p>我们编写代码给出代价函数及其偏导数然后传入梯度下降算法中，接下来算法则会为我们最小化代价函数给出参数的最优解。这类算法被称为<strong>最优化算法(Optimization Algorithms)</strong>，梯度下降算法不是唯一的最小化算法<sup><a href='#dfref-footnote-1' name='ref-footnote-1'>1</a></sup>。</p><p>一些最优化算法：</p><ul><li><p>梯度下降法(Gradient Descent)</p></li><li><p>共轭梯度算法(Conjugate gradient)</p></li><li><p>牛顿法和拟牛顿法(Newton&#39;s method &amp; Quasi-Newton Methods)</p><ul><li>DFP算法</li><li>局部优化法(BFGS)</li><li>有限内存局部优化法(L-BFGS)</li></ul></li><li><p>拉格朗日乘数法(Lagrange multiplier)</p></li></ul><p>比较梯度下降算法：一些最优化算法虽然会更为复杂，难以调试，自行实现又困难重重，开源库又效率也不一，哎，做个调包侠还得碰运气。不过这些算法通常效率更高，并无需选择学习速率 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-72-Frame" tabindex="-1" style="font-size: 100%; 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type="math/tex" id="MathJax-Element-72">\alpha</script>（少一个参数少一份痛苦啊！）。</p><p>Octave/Matlab 中对这类高级算法做了封装，易于调用。</p><p>&nbsp;</p><p>假设有 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-250-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="28.332ex" height="2.811ex" viewBox="0 -906.7 12198.5 1210.2" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E260-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E260-MJMAIN-28" d="M94 250Q94 319 104 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMAIN-35" x="5751" y="0"></use><g transform="translate(6252,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMAIN-32" x="550" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMAIN-2B" x="7318" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMAIN-28" x="8318" y="0"></use><g transform="translate(8708,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMAIN-32" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMAIN-2212" x="9853" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMAIN-35" x="10854" y="0"></use><g transform="translate(11355,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMAIN-32" x="550" y="513"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-250">J(\theta) = (\theta_1-5)^2 + (\theta_2-5)^2</script>，要求参数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-251-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.53ex" height="5.846ex" viewBox="0 -1509.8 4103 2517" role="img" focusable="false" style="vertical-align: -2.339ex;"><defs><path stroke-width="1" id="E261-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 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409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E261-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path stroke-width="1" id="E261-MJSZ3-5B" d="M247 -949V1450H516V1388H309V-887H516V-949H247Z"></path><path stroke-width="1" id="E261-MJSZ3-5D" d="M11 1388V1450H280V-949H11V-887H218V1388H11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E261-MJMATHI-3B8" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E261-MJMAIN-3D" x="747" y="0"></use><g transform="translate(1803,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E261-MJSZ3-5B"></use><g transform="translate(695,0)"><g transform="translate(-15,0)"><g transform="translate(0,650)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E261-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E261-MJMAIN-31" x="663" y="-213"></use></g><g transform="translate(0,-750)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E261-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E261-MJMAIN-32" x="663" y="-213"></use></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E261-MJSZ3-5D" x="1770" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-251">\theta=\begin{bmatrix} \theta_1\\\theta_2\end{bmatrix}</script>的最优值。</p><p>下面为 Octave/Matlab 求解最优化问题的代码实例：</p><ol start='' ><li>创建一个函数以返回代价函数及其偏导数：</li></ol><pre class="md-fences md-end-block md-fences-with-lineno" lang="matlab"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>10</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">function</span> [<span class="cm-variable">jVal</span>, <span class="cm-variable">gradient</span>] = <span class="cm-variable">costFunction</span>(<span class="cm-variable">theta</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-comment">% code to compute J(theta)</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-variable">jVal</span>=(<span class="cm-variable">theta</span>(<span class="cm-number">1</span>)<span class="cm-number">-5</span>)<span class="cm-operator">^</span><span class="cm-number">2</span><span class="cm-operator">+</span>(<span class="cm-variable">theta</span>(<span class="cm-number">2</span>)<span class="cm-number">-5</span>)<span class="cm-operator">^</span><span class="cm-number">2</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-comment">% code to compute derivative of J(theta)</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-variable">gradient</span>=<span class="cm-builtin">zeros</span>(<span class="cm-number">2</span>,<span class="cm-number">1</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-variable">gradient</span>(<span class="cm-number">1</span>)=<span class="cm-number">2</span><span class="cm-operator">*</span>(<span class="cm-variable">theta</span>(<span class="cm-number">1</span>)<span class="cm-number">-5</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-variable">gradient</span>(<span class="cm-number">2</span>)=<span class="cm-number">2</span><span class="cm-operator">*</span>(<span class="cm-variable">theta</span>(<span class="cm-number">2</span>)<span class="cm-number">-5</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 224px;"></div><div class="CodeMirror-gutters" style="height: 254px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><ol start='2' ><li>将 <code>costFunction</code> 函数及所需参数传入最优化函数 <code>fminunc</code>，以求解最优化问题：</li></ol><pre class="md-fences md-end-block md-fences-with-lineno" lang="matlab"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 31px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 27px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>3</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -27px; width: 27px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 18px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">options</span> = <span class="cm-variable">optimset</span>(<span class="cm-string">'GradObj'</span>, <span class="cm-string">'on'</span>, <span class="cm-string">'MaxIter'</span>, <span class="cm-number">100</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">initialTheta</span> = <span class="cm-builtin">zeros</span>(<span class="cm-number">2</span>,<span class="cm-number">1</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 18px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; [<span class="cm-variable">optTheta</span>, <span class="cm-variable">functionVal</span>, <span class="cm-variable">exitFlag</span>] = <span class="cm-variable">fminunc</span>(<span class="cm-operator">@</span><span class="cm-variable">costFunction</span>, <span class="cm-variable">initialTheta</span>, <span class="cm-variable">options</span>);</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 67px;"></div><div class="CodeMirror-gutters" style="height: 97px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 26px;"></div></div></div></div></pre><blockquote><p><code>&#39;GradObj&#39;, &#39;on&#39;</code>: 启用梯度目标参数（则需要将梯度传入算法）</p><p><code>&#39;MaxIter&#39;, 100</code>: 最大迭代次数为 100 次</p><p><code>@xxx</code>: Octave/Matlab 中的函数指针</p><p><code>optTheta</code>: 最优化得到的参数向量</p><p><code>functionVal</code>: 引用函数最后一次的返回值</p><p><code>exitFlag</code>: 标记代价函数是否收敛</p></blockquote><p>注：Octave/Matlab 中可以使用 <code>help fminunc</code> 命令随时查看函数的帮助文档。</p><ol start='3' ><li>返回结果</li></ol><pre class="md-fences md-end-block md-fences-with-lineno" lang=""> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 31px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 27px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>8</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -27px; width: 27px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 18px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">optTheta =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">functionVal = 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 18px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">exitFlag = 1</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 179px;"></div><div class="CodeMirror-gutters" style="height: 209px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 26px;"></div></div></div></div></pre><h2><a name='header-n271' class='md-header-anchor '></a>6.7 多类别分类: 一对多(Multiclass Classification: One-vs-all)</h2><p>一直在讨论二元分类问题，这里谈谈多类别分类问题（比如天气预报）。</p><p><img src='image/20180112_001720.png' alt='' /></p><p>原理是，转化多类别分类问题为<strong>多个二元分类问题</strong>，这种方法被称为 One-vs-all。</p><p>正式定义：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-252-Frame" tabindex="-1" style="font-size: 100%; display: 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66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E143-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">k</script>: 类别总数，如上图 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-255-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.472ex" height="1.994ex" viewBox="0 -755.9 2356.1 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path 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id="MathJax-Element-144">k</script> 维向量。</p><p>对于某个样本实例，需计算所有的 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.211ex" height="1.994ex" viewBox="0 -755.9 521.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E143-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E269-MJMATHI-3B8" x="815" y="-474"></use></g><g transform="translate(5665,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E269-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E269-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E269-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-259">y = \mathop{\max}\limits_i\,h_\theta^{\left( i \right)}\left( x \right)</script>。</p><h1><a name='header-n289' class='md-header-anchor '></a>7 正则化(Regularization)</h1><h2><a name='header-n290' class='md-header-anchor '></a>7.1 过拟合问题(The Problem of Overfitting)</h2><p>对于拟合的表现，可以分为三类情况：</p><ul><li><p><strong>欠拟合(Underfitting)</strong></p><p>无法很好的拟合训练集中的数据，预测值和实际值的误差很大，这类情况被称为欠拟合。拟合模型比较简单（特征选少了）时易出现这类情况。类似于，你上课不好好听，啥都不会，下课也差不多啥都不会。</p></li><li><p><strong>优良的拟合(Just right)</strong></p><p>不论是训练集数据还是不在训练集中的预测数据，都能给出较为正确的结果。类似于，学霸学神！</p></li><li><p><strong>过拟合(Overfitting)</strong></p><p>能很好甚至完美拟合训练集中的数据，即 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-260-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.148ex" height="2.577ex" viewBox="0 -806.1 3938.6 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E270-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E270-MJMAIN-28" d="M94 250Q94 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E270-MJMAIN-28" x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E270-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E270-MJMAIN-29" x="1492" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E270-MJMAIN-2192" x="2159" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E270-MJMAIN-30" x="3438" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-260">J(\theta) \to 0</script>，但是对于不在训练集中的<strong>新数据</strong>，预测值和实际值的误差会很大，<strong>泛化能力弱</strong>，这类情况被称为过拟合。拟合模型过于复杂（特征选多了）时易出现这类情况。类似于，你上课跟着老师做题都会都听懂了，下课遇到新题就懵了不会拓展。</p></li></ul><p>线性模型中的拟合情况(左图欠拟合，右图过拟合)：
+<div  id='write'  class = 'is-node show-fences-line-number'><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n2">6 逻辑回归(Logistic Regression)</a></span><span class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n3">6.1 分类(Classification)</a></span><span class="md-toc-item md-toc-h2" data-ref="n40"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n40">6.2 假设函数表示(Hypothesis Representation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n64"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n64">6.3 决策边界(Decision Boundary)</a></span><span class="md-toc-item md-toc-h2" data-ref="n109"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n109">6.4 代价函数(Cost Function)</a></span><span class="md-toc-item md-toc-h2" data-ref="n138"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n138">6.5 简化的成本函数和梯度下降(Simplified Cost Function and Gradient Descent)</a></span><span class="md-toc-item md-toc-h2" data-ref="n200"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n200">6.6 进阶优化(Advanced Optimization)</a></span><span class="md-toc-item md-toc-h2" data-ref="n270"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n270">6.7 多类别分类: 一对多(Multiclass Classification: One-vs-all)</a></span><span class="md-toc-item md-toc-h1" data-ref="n288"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n288">7 正则化(Regularization)</a></span><span class="md-toc-item md-toc-h2" data-ref="n289"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n289">7.1 过拟合问题(The Problem of Overfitting)</a></span><span class="md-toc-item md-toc-h2" data-ref="n365"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n365">7.2 代价函数(Cost Function)</a></span><span class="md-toc-item md-toc-h2" data-ref="n422"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n422">7.3 线性回归正则化(Regularized Linear Regression)</a></span><span class="md-toc-item md-toc-h2" data-ref="n452"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n452">7.4 逻辑回归正则化(Regularized Logistic Regression)</a></span></p></div><h1><a name='header-n2' class='md-header-anchor '></a>6 逻辑回归(Logistic Regression)</h1><h2><a name='header-n3' class='md-header-anchor '></a>6.1 分类(Classification)</h2><p>在分类问题中，预测的结果是离散值（结果是否属于某一类），逻辑回归算法(Logistic Regression)被用于解决这类分类问题。</p><ul><li>垃圾邮件判断</li><li>金融欺诈判断</li><li>肿瘤诊断</li></ul><p>讨论肿瘤诊断问题：</p><p><img src='image/20180109_144040.png' alt='' /></p><p>肿瘤诊断问题的目的是告诉病人<strong>是否</strong>为恶性肿瘤，是一个<strong>二元分类问题(binary class problems)</strong>，则定义 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-61-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.68ex" height="2.577ex" 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xlink:href="#E63-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E63-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E63-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E63-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E63-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E63-MJAMS-2A7E" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E63-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E63-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E63-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-62">h_\theta(x) \geqslant 0.5</script> ，预测为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-121-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E122-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 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xlink:href="#E122-MJMAIN-31" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-121">y = 1</script>，即正向类；</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-64-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.551ex" height="2.577ex" viewBox="0 -806.1 4973.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E65-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 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328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E65-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E65-MJMAIN-3C" d="M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 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396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E65-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E65-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E65-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E65-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E65-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E65-MJMAIN-3C" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E65-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E65-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E65-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-64">h_\theta(x) \lt 0.5</script> ，预测为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-119-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E120-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E120-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E120-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-30" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-119">y = 0</script>，即负向类。</p><p>即以 0.5 为<strong>阈值</strong>(threshold)，则我们就可以根据线性回归结果，得到相对正确的分类结果 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-66-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.155ex" height="1.877ex" viewBox="0 -504.6 497.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E67-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E67-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-66">y</script>。</p><p>&nbsp;</p><p>接下来加入偏差项，线性回归算法给出了靛青色的拟合直线，如果阈值仍然为 0.5，可以看到算法在某些情况下会给出完全错误的结果，对于癌症、肿瘤诊断这类要求预测极其精确的问题，这种情况是无法容忍的。</p><p>不仅如此，线性回归算法的值域为全体实数集（<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-67-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.086ex" height="2.577ex" viewBox="0 -806.1 4342.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E68-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E68-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E68-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E68-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 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501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path stroke-width="1" id="E68-MJMATHI-52" d="M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 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y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-67">h_\theta(x) \in R</script>），则当线性回归函数给出诸如 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-68-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="31.626ex" height="2.577ex" viewBox="0 -806.1 13616.8 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E69-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-3D" x="9279" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-2212" x="10335" y="0"></use><g transform="translate(11114,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-31"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-30" x="1001" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-30" x="1501" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-30" x="2002" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-68">h_\theta(x) = 10000, h_\theta(x) = -10000</script> 等很大/很小(负数)的数值时，结果 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-69-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.68ex" height="2.577ex" viewBox="0 -806.1 4167.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E70-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E70-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path stroke-width="1" id="E70-MJMAIN-7B" d="M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 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x="1720" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMAIN-30" x="2221" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMAIN-2C" x="2721" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMAIN-31" x="3166" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMAIN-7D" x="3667" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-69">y \in \lbrace 0, 1\rbrace</script>，这显得非常怪异。</p><p>&nbsp;</p><p>区别于线性回归算法，逻辑回归算法是一个分类算法，<strong>其输出值永远在 0 到 1 之间</strong>，即 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-70-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.49ex" height="2.577ex" viewBox="0 -806.1 5808.2 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E71-MJMATHI-68" d="M137 683Q138 683 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class='md-header-anchor '></a>6.2 假设函数表示(Hypothesis Representation)</h2><p>为了使 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-71-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.49ex" height="2.577ex" viewBox="0 -806.1 5808.2 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E72-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 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xlink:href="#E73-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-3D" x="2804" y="0"></use><g transform="translate(3860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-78" x="4928" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-72">h_\theta \left( x \right)=\theta^{T}x</script>，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-73-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.116ex" height="1.877ex" viewBox="0 -504.6 480.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E74-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E74-MJMATHI-67" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-73">g</script> 表示逻辑函数(<a href='https://en.wikipedia.org/wiki/Logistic_function' target='_blank'>logistic function</a>)，复合起来，则称为逻辑回归函数。</p><p>逻辑函数是 S 形函数，会将所有实数映射到 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-74-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.168ex" height="2.577ex" viewBox="0 -806.1 2225.2 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E75-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path 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d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E75-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E75-MJMAIN-30" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E75-MJMAIN-2C" x="890" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E75-MJMAIN-31" x="1335" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E75-MJMAIN-29" x="1835" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-74">(0, 1)</script> 范围。</p><p><a href='https://en.wikipedia.org/wiki/Sigmoid_function' target='_blank'>sigmoid 函数</a>（如下图）是逻辑函数的特殊情况，其公式为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-75-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.814ex" height="3.395ex" viewBox="0 -956.9 5517.2 1461.5" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E76-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 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transform="translate(329,204)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E77-MJMAIN-2212" x="0" y="0"></use><g transform="translate(389,0)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E77-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E77-MJMATHI-54" x="469" y="319"></use></g><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E77-MJMATHI-78" x="2052" y="0"></use></g></g></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-76">h_{\theta}(x)=g(\theta^Tx) =\frac{1}{1+e^{-\theta^Tx}}</script></p><p>逻辑回归模型中，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-77-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.868ex" height="2.577ex" viewBox="0 -806.1 2526.7 1109.7" role="img" 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xlink:href="#E78-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-77">h_\theta \left( x \right)</script> 的作用是，根据输入 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-78-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.33ex" height="1.41ex" viewBox="0 -504.6 572.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E79-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E79-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-78">x</script> 以及参数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-151-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E152-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-151">\theta</script>，计算得出”输出 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-112-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E113-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E113-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E113-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 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xlink:href="#E85-MJMATHI-67" x="3860" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E85-MJMAIN-28" x="4341" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E85-MJMATHI-7A" x="4730" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E85-MJMAIN-29" x="5199" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E85-MJMAIN-3D" x="5866" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E85-MJMATHI-67" x="6922" y="0"></use><g transform="translate(7569,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E85-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E85-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E85-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E85-MJMATHI-78" x="1526" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E85-MJSZ1-29" x="2098" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-84">h_\theta \left( x \right)=g(z)=g\left(\theta^{T}x \right)</script>。</p><p>为了得出分类的结果，这里和前面一样，规定以 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-85-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="1.994ex" viewBox="0 -755.9 1279.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E86-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E86-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="1" id="E86-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E86-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E86-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E86-MJMAIN-35" x="779" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-85">0.5</script> 为阈值：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-86-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="21.323ex" height="5.846ex" viewBox="0 -1509.8 9180.7 2517" role="img" focusable="false" style="vertical-align: -2.339ex;"><defs><path stroke-width="1" id="E87-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E87-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E87-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E87-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E87-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E87-MJMAIN-2265" d="M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 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666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path stroke-width="1" id="E87-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path><path stroke-width="1" id="E87-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 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xlink:href="#E87-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-3C" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-35" x="779" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-2192" x="5251" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMATHI-79" x="6529" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-3D" x="7304" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-30" x="8361" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-86">\begin{align*}& h_\theta(x) \geq 0.5 \rightarrow y = 1 \newline& h_\theta(x) < 0.5 \rightarrow y = 0 \newline\end{align*}</script></p><p>回忆一下 sigmoid 函数的图像：</p><p><img src='image/2413fbec8ff9fa1f19aaf78265b8a33b_Logistic_function.png' alt='sigmoid function' /></p><p>观察可得当 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-87-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.084ex" height="2.577ex" viewBox="0 -806.1 4341.6 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E88-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path stroke-width="1" id="E88-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E88-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E88-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E88-MJMAIN-2265" d="M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 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0</script>。</p><p>同线性回归模型的不同点在于： <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-90-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="31.512ex" height="5.846ex" viewBox="0 -1509.8 13567.4 2517" role="img" focusable="false" style="vertical-align: -2.339ex;"><defs><path stroke-width="1" id="E91-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMAIN-34" x="663" y="-213"></use></g><g transform="translate(13046,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMAIN-32" x="809" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMAIN-32" x="809" y="-435"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJSZ1-29" x="14072" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-99">{h_\theta}\left( x \right)=g\left( {\theta_0}+{\theta_1}{x_1}+{\theta_{2}}{x_{2}}+{\theta_{3}}x_{1}^{2}+{\theta_{4}}x_{2}^{2} \right)</script></p><p>这里取 <span class="MathJax_Preview"></span><span class="MathJax_SVG" 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transform="translate(0,-4843.8708609271525) scale(1,5.129139072847682)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJSZ4-23A2"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJSZ4-23A3" x="0" y="-5955"></use></g><g transform="translate(834,0)"><g transform="translate(-15,0)"><g transform="translate(0,2750)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMAIN-2212" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMAIN-31" x="778" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMAIN-30" x="389" y="1350"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMAIN-30" x="389" y="-50"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMAIN-31" x="389" y="-1450"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMAIN-31" x="389" y="-2850"></use></g></g><g transform="translate(2265,3550)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJSZ4-23A4" x="0" y="-1155"></use><g transform="translate(0,-4843.8708609271525) scale(1,5.129139072847682)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJSZ4-23A5"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJSZ4-23A6" x="0" y="-5955"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-100">\theta = \begin{bmatrix} -1\\0\\0\\1\\1\end{bmatrix}</script>，决策边界对应了一个在原点处的单位圆（<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-101-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.978ex" height="2.577ex" viewBox="0 -906.7 6018.1 1109.7" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E102-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 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x="5517" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-101">{x_1}^2+{x_2}^2 = 1</script>），如此便可给出分类结果，如图中品红色曲线：</p><p>&nbsp;</p><p><img src='image/20180111_000653.png' alt='' /></p><p>当然，通过一些更为复杂的多项式，还能拟合那些图像显得非常怪异的数据，使得决策边界形似碗状、爱心状等等。</p><p>&nbsp;</p><p>简单来说，决策边界就是<strong>分类的分界线</strong>，分类现在实际就由 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-102-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.088ex" height="1.41ex" viewBox="0 -504.6 468.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E103-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMATHI-7A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-102">z</script> (中的 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-151-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E152-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-151">\theta</script>)决定啦。</p><h2><a name='header-n109' class='md-header-anchor '></a>6.4 代价函数(Cost Function)</h2><p>那我们怎么知道决策边界是啥样？<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-151-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E152-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-151">\theta</script> 多少时能很好的拟合数据？当然，见招拆招，总要来个 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-153-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.371ex" height="2.577ex" viewBox="0 -806.1 1882 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E154-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E154-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" 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x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-153">J(\theta)</script>。</p><p>如果直接套用线性回归的代价函数： <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-106-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="31.932ex" height="5.846ex" viewBox="0 -1459.5 13748.3 2517" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="1" id="E107-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 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406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E107-MJSZ1-28" d="M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 56T152 251Z"></path><path stroke-width="1" id="E107-MJSZ1-29" d="M305 251Q305 -145 69 -349H56Q43 -349 39 -347T35 -338Q37 -333 60 -307T108 -239T160 -136T204 27T221 250T204 473T160 636T108 740T60 807T35 839Q35 850 50 850H56H69Q197 743 256 566Q305 425 305 251Z"></path><path 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMATHI-6D" x="500" y="0"></use></g></g></g><g transform="translate(4884,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJSZ1-2211" x="46" y="0"></use><g transform="translate(0,-890)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-3D" x="345" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-31" x="1124" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMATHI-6D" x="372" y="1344"></use></g><g transform="translate(6199,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMATHI-3B8" x="815" y="-219"></use></g><g transform="translate(1633,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJSZ1-29" x="1926" y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-2212" x="4240" y="0"></use><g transform="translate(5241,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMATHI-79" x="0" y="0"></use><g transform="translate(499,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJSZ1-29" x="6636" y="-1"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-32" x="10033" y="878"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-106">J\left( {\theta} \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( h_{\theta} \left({x}^{\left( i \right)} \right)-{y}^{\left( i \right)} \right)}^{2}}}</script></p><p>其中 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y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E130-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E130-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E130-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E130-MJMAIN-3D" x="2637" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E130-MJMATHI-67" x="3694" y="0"></use><g transform="translate(4341,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E130-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E130-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E130-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E130-MJMATHI-78" x="1526" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E130-MJSZ1-29" x="2098" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-129">h_\theta(x) = g\left(\theta^{T}x \right)</script>，可绘制关于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-153-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.371ex" height="2.577ex" viewBox="0 -806.1 1882 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E154-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E154-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E154-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E154-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMATHI-4A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMAIN-28" x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-153">J(\theta)</script> 的图像，如下图</p><p><img src='image/20180111_080314.png' alt='' /></p><p>回忆线性回归中的平方损失函数，其是一个二次凸函数（碗状），二次凸函数的重要性质是只有一个局部最小点即全局最小点。上图中有许多局部最小点，这样将使得梯度下降算法无法确定收敛点是全局最优。</p><p><img src='image/20180111_080514.png' alt='' /></p><p>如果此处的损失函数也是一个凸函数，是否也有同样的性质，从而最优化？这类讨论凸函数最优值的问题，被称为<strong>凸优化问题(Convex optimization)</strong>。</p><p>当然，损失函数不止平方损失函数一种。</p><p>对于逻辑回归，更换平方损失函数为<strong>对数损失函数</strong>，可由统计学中的最大似然估计方法推出代价函数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-153-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.371ex" height="2.577ex" viewBox="0 -806.1 1882 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E154-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E154-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 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xlink:href="#E111-MJMAIN-30" x="2642" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-110">\begin{align*}& J(\theta) = \dfrac{1}{m} \sum_{i=1}^m \mathrm{Cost}(h_\theta(x^{(i)}),y^{(i)}) \newline & \mathrm{Cost}(h_\theta(x),y) = -\log(h_\theta(x)) \; & \text{if y = 1} \newline & \mathrm{Cost}(h_\theta(x),y) = -\log(1-h_\theta(x)) \; & \text{if y = 0}\end{align*}</script></p><p>则有关于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-153-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.371ex" height="2.577ex" viewBox="0 -806.1 1882 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E154-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E154-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E154-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E154-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMATHI-4A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMAIN-28" x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-153">J(\theta)</script> 的图像如下：</p><p><img src='image/20180111_080614.png' alt='' /></p><p>如左图，当训练集的结果为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-112-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E113-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E113-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E113-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E113-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E113-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E113-MJMAIN-31" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-112">y=1</script>（正样本）时，随着假设函数趋向于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-113-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.162ex" height="1.994ex" viewBox="0 -755.9 500.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E114-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E114-MJMAIN-31" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-113">1</script>，代价函数的值会趋于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-191-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.162ex" height="1.994ex" viewBox="0 -755.9 500.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E193-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E193-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-191">0</script>，即意味着拟合程度很好。如果假设函数此时趋于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-191-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.162ex" height="1.994ex" viewBox="0 -755.9 500.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E193-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E193-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-191">0</script>，则会给出一个<strong>很高的代价</strong>，拟合程度<strong>差</strong>，算法会根据其迅速纠正 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-151-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E152-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-151">\theta</script> 值，右图 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-117-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E118-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E118-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E118-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E118-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E118-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E118-MJMAIN-30" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-117">y=0</script> 同理。</p><p>区别于平方损失函数，对数损失函数也是一个凸函数，但没有局部最优值。</p><h2><a name='header-n138' class='md-header-anchor '></a>6.5 简化的成本函数和梯度下降(Simplified Cost Function and Gradient Descent)</h2><p>由于懒得分类讨论，对于二元分类问题，我们可把代价函数<strong>简化</strong>为一个函数： 
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130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E119-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E119-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E119-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E119-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 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xlink:href="#E119-MJMAIN-29" x="2916" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMAIN-2212" x="15450" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMAIN-28" x="16451" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMAIN-31" x="16840" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMAIN-2212" x="17563" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMATHI-79" x="18564" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMAIN-29" x="19061" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMAIN-D7" x="19673" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMATHI-6C" x="20674" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMATHI-6F" x="20972" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMATHI-67" x="21458" y="0"></use><g transform="translate(22105,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMAIN-31" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMAIN-2212" x="1112" y="0"></use><g transform="translate(2112,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMATHI-3B8" x="815" y="-219"></use></g><g transform="translate(3288,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E119-MJMAIN-29" x="4639" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-118">Cost\left( {h_\theta}\left( x \right),y \right)=-y\times log\left( {h_\theta}\left( x \right) \right)-(1-y)\times log\left( 1-{h_\theta}\left( x \right) \right)</script></p><p>当 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-119-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E120-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 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320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E193-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-191">0</script>，当 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-121-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E122-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 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y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E130-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E130-MJMATHI-78" x="1526" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E130-MJSZ1-29" x="2098" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-129">h_\theta(x) = g\left(\theta^{T}x \right)</script>，不过求导后的结果也相同。</p><p>向量化实现：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-130-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="26.233ex" height="3.161ex" viewBox="0 -906.7 11294.5 1361" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="1" id="E131-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 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x="458" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMAIN-2212" x="1181" y="0"></use><g transform="translate(2181,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMATHI-79" x="0" y="0"></use><g transform="translate(499,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJSZ1-29" x="3576" y="-1"></use></g><g transform="translate(15875,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMAIN-6C"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMAIN-6F" x="278" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMAIN-67" x="779" y="0"></use></g><g transform="translate(17155,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJSZ1-28"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMAIN-31" x="458" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMAIN-2212" x="1181" y="0"></use><g transform="translate(2181,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMATHI-3B8" x="815" y="-219"></use></g><g transform="translate(3357,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJSZ1-29" x="1926" y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E133-MJSZ1-29" x="5741" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-132">f(\theta) = {{y}^{(i)}}\log \left( {h_\theta}\left( {{x}^{(i)}} \right) \right)+\left( 1-{{y}^{(i)}} \right)\log \left( 1-{h_\theta}\left( {{x}^{(i)}} \right) \right)</script></p><p>忆及 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-634-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.793ex" height="2.461ex" viewBox="0 -956.9 4646.9 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E922-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E922-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E922-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="1" id="E922-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E922-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E922-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E922-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E922-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E922-MJMAIN-29" x="734" y="0"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-634">z=\Theta^Tx^{(i)}</script>，将 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-635-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="26.108ex" height="4.095ex" viewBox="0 -956.9 11241 1763.1" role="img" focusable="false" style="vertical-align: -1.872ex;"><defs><path stroke-width="1" id="E924-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 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xlink:href="#E924-MJMATHI-67" x="4589" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMAIN-28" x="5069" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMATHI-7A" x="5459" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMAIN-29" x="5927" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMAIN-3D" x="6594" y="0"></use><g transform="translate(7373,0)"><g transform="translate(397,0)"><rect stroke="none" width="3349" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMAIN-31" x="2118" y="572"></use><g transform="translate(60,-664)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMAIN-2B" x="500" y="0"></use><g transform="translate(904,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMATHI-65" x="0" y="0"></use><g transform="translate(329,204)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMAIN-2212" x="0" y="0"></use><g transform="translate(389,0)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMATHI-54" x="469" y="319"></use></g><g transform="translate(1026,0)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMATHI-78" x="0" y="0"></use><g transform="translate(286,179)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E924-MJMAIN-29" x="735" y="0"></use></g></g></g></g></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-635">h_\theta(x^{(i)}) = g(z) =\frac{1}{1+{{e}^{-{\theta^T}{{x}^{(i)}}}}} </script> 带入得</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-134-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="55.675ex" height="4.679ex" viewBox="0 -1208.2 23971.2 2014.4" role="img" focusable="false" style="vertical-align: -1.872ex;"><defs><path stroke-width="1" id="E135-MJMATHI-66" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path stroke-width="1" id="E135-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E135-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E831-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E831-MJSZ1-29" x="2993" y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E831-MJMAIN-3D" x="7804" y="0"></use><g transform="translate(8860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E831-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,521)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E831-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E831-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E831-MJMAIN-29" x="734" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E831-MJMATHI-6A" x="809" y="-430"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-591">\frac{\partial z}{\partial {\theta_{j}}}=\frac{\partial }{\partial {\theta_{j}}}\left( \theta^Tx^{(i)}  \right)=x^{(i)}_j</script></p><p>所以有：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-831-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="56.374ex" height="3.978ex" viewBox="0 -1007.2 24271.9 1712.8" role="img" focusable="false" style="vertical-align: -1.639ex;"><defs><path stroke-width="1" id="E1294-MJMAIN-2202" d="M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 372 401 400T333 428Q270 428 222 382Q197 354 183 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10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E152-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-151">\theta</script> 的最优值，在使用梯度下降算法时，如果不需要观察代价函数的收敛情况，则直接计算 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-153-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.371ex" height="2.577ex" viewBox="0 -806.1 1882 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E154-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E154-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E154-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 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id="MathJax-Element-153">J(\theta)</script> 的导数项即可，而不需要计算 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-153-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.371ex" height="2.577ex" viewBox="0 -806.1 1882 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E154-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E154-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E154-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E154-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMATHI-4A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMAIN-28" x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E154-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-153">J(\theta)</script> 值。</p><p>我们编写代码给出代价函数及其偏导数然后传入梯度下降算法中，接下来算法则会为我们最小化代价函数给出参数的最优解。这类算法被称为<strong>最优化算法(Optimization Algorithms)</strong>，梯度下降算法不是唯一的最小化算法<sup><a href='#dfref-footnote-1' name='ref-footnote-1'>1</a></sup>。</p><p>一些最优化算法：</p><ul><li><p>梯度下降法(Gradient Descent)</p></li><li><p>共轭梯度算法(Conjugate gradient)</p></li><li><p>牛顿法和拟牛顿法(Newton&#39;s method &amp; Quasi-Newton Methods)</p><ul><li>DFP算法</li><li>局部优化法(BFGS)</li><li>有限内存局部优化法(L-BFGS)</li></ul></li><li><p>拉格朗日乘数法(Lagrange multiplier)</p></li></ul><p>比较梯度下降算法：一些最优化算法虽然会更为复杂，难以调试，自行实现又困难重重，开源库又效率也不一，哎，做个调包侠还得碰运气。不过这些算法通常效率更高，并无需选择学习速率 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-154-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.41ex" viewBox="0 -504.6 640.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E155-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 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id="MathJax-Element-156">\theta=\begin{bmatrix} \theta_1\\\theta_2\end{bmatrix}</script>的最优值。</p><p>下面为 Octave/Matlab 求解最优化问题的代码实例：</p><ol start='' ><li>创建一个函数以返回代价函数及其偏导数：</li></ol><pre class="md-fences md-end-block md-fences-with-lineno" lang="matlab"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>10</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">function</span> [<span class="cm-variable">jVal</span>, <span class="cm-variable">gradient</span>] = <span class="cm-variable">costFunction</span>(<span class="cm-variable">theta</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-comment">% code to compute J(theta)</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-variable">jVal</span>=(<span class="cm-variable">theta</span>(<span class="cm-number">1</span>)<span class="cm-number">-5</span>)<span class="cm-operator">^</span><span class="cm-number">2</span><span class="cm-operator">+</span>(<span class="cm-variable">theta</span>(<span class="cm-number">2</span>)<span class="cm-number">-5</span>)<span class="cm-operator">^</span><span class="cm-number">2</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-comment">% code to compute derivative of J(theta)</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-variable">gradient</span>=<span class="cm-builtin">zeros</span>(<span class="cm-number">2</span>,<span class="cm-number">1</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-variable">gradient</span>(<span class="cm-number">1</span>)=<span class="cm-number">2</span><span class="cm-operator">*</span>(<span class="cm-variable">theta</span>(<span class="cm-number">1</span>)<span class="cm-number">-5</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-variable">gradient</span>(<span class="cm-number">2</span>)=<span class="cm-number">2</span><span class="cm-operator">*</span>(<span class="cm-variable">theta</span>(<span class="cm-number">2</span>)<span class="cm-number">-5</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 224px;"></div><div class="CodeMirror-gutters" style="height: 254px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><ol start='2' ><li>将 <code>costFunction</code> 函数及所需参数传入最优化函数 <code>fminunc</code>，以求解最优化问题：</li></ol><pre class="md-fences md-end-block md-fences-with-lineno" lang="matlab"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 31px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 27px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>3</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -27px; width: 27px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 18px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">options</span> = <span class="cm-variable">optimset</span>(<span class="cm-string">'GradObj'</span>, <span class="cm-string">'on'</span>, <span class="cm-string">'MaxIter'</span>, <span class="cm-number">100</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">initialTheta</span> = <span class="cm-builtin">zeros</span>(<span class="cm-number">2</span>,<span class="cm-number">1</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 18px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; [<span class="cm-variable">optTheta</span>, <span class="cm-variable">functionVal</span>, <span class="cm-variable">exitFlag</span>] = <span class="cm-variable">fminunc</span>(<span class="cm-operator">@</span><span class="cm-variable">costFunction</span>, <span class="cm-variable">initialTheta</span>, <span class="cm-variable">options</span>);</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 67px;"></div><div class="CodeMirror-gutters" style="height: 97px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 26px;"></div></div></div></div></pre><blockquote><p><code>&#39;GradObj&#39;, &#39;on&#39;</code>: 启用梯度目标参数（则需要将梯度传入算法）</p><p><code>&#39;MaxIter&#39;, 100</code>: 最大迭代次数为 100 次</p><p><code>@xxx</code>: Octave/Matlab 中的函数指针</p><p><code>optTheta</code>: 最优化得到的参数向量</p><p><code>functionVal</code>: 引用函数最后一次的返回值</p><p><code>exitFlag</code>: 标记代价函数是否收敛</p></blockquote><p>注：Octave/Matlab 中可以使用 <code>help fminunc</code> 命令随时查看函数的帮助文档。</p><ol start='3' ><li>返回结果</li></ol><pre class="md-fences md-end-block md-fences-with-lineno" lang=""> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 31px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 27px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>8</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -27px; width: 27px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 18px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">optTheta =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">functionVal = 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 18px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -27px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 18px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">exitFlag = 1</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 179px;"></div><div class="CodeMirror-gutters" style="height: 209px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 26px;"></div></div></div></div></pre><h2><a name='header-n270' class='md-header-anchor '></a>6.7 多类别分类: 一对多(Multiclass Classification: One-vs-all)</h2><p>一直在讨论二元分类问题，这里谈谈多类别分类问题（比如天气预报）。</p><p><img src='image/20180112_001720.png' alt='' /></p><p>原理是，转化多类别分类问题为<strong>多个二元分类问题</strong>，这种方法被称为 One-vs-all。</p><p>正式定义：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-157-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="39.051ex" height="3.511ex" viewBox="0 -1107.7 16813.5 1511.8" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E158-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 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id="MathJax-Element-16-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.211ex" height="1.994ex" viewBox="0 -755.9 521.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E16-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 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153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E164-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E164-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 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252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E16-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-16">k</script> 种分类情况得到 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-163-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.481ex" height="2.577ex" viewBox="0 -806.1 2360 1109.7" 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E166-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E166-MJMAIN-29" x="734" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E166-MJMATHI-3B8" x="815" y="-474"></use></g><g transform="translate(5665,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E166-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E166-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E166-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-164">y = \mathop{\max}\limits_i\,h_\theta^{\left( i \right)}\left( x \right)</script>。</p><h1><a name='header-n288' class='md-header-anchor '></a>7 正则化(Regularization)</h1><h2><a name='header-n289' class='md-header-anchor '></a>7.1 过拟合问题(The Problem of Overfitting)</h2><p>对于拟合的表现，可以分为三类情况：</p><ul><li><p><strong>欠拟合(Underfitting)</strong></p><p>无法很好的拟合训练集中的数据，预测值和实际值的误差很大，这类情况被称为欠拟合。拟合模型比较简单（特征选少了）时易出现这类情况。类似于，你上课不好好听，啥都不会，下课也差不多啥都不会。</p></li><li><p><strong>优良的拟合(Just right)</strong></p><p>不论是训练集数据还是不在训练集中的预测数据，都能给出较为正确的结果。类似于，学霸学神！</p></li><li><p><strong>过拟合(Overfitting)</strong></p><p>能很好甚至完美拟合训练集中的数据，即 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-165-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.148ex" height="2.577ex" viewBox="0 -806.1 3938.6 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E167-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 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stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E167-MJMATHI-4A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E167-MJMAIN-28" x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E167-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E167-MJMAIN-29" x="1492" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E167-MJMAIN-2192" x="2159" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E167-MJMAIN-30" x="3438" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-165">J(\theta) \to 0</script>，但是对于不在训练集中的<strong>新数据</strong>，预测值和实际值的误差会很大，<strong>泛化能力弱</strong>，这类情况被称为过拟合。拟合模型过于复杂（特征选多了）时易出现这类情况。类似于，你上课跟着老师做题都会都听懂了，下课遇到新题就懵了不会拓展。</p></li></ul><p>线性模型中的拟合情况(左图欠拟合，右图过拟合)：
 <img src='image/20180112_091654.png' alt='' /></p><p>逻辑分类模型中的拟合情况：
 <img src='image/20180112_092027.png' alt='' /></p><p>&nbsp;</p><p>为了度量拟合表现，引入：</p><ul><li><p>偏差(bias)</p><p>指模型的预测值与真实值的<strong>偏离程度</strong>。偏差越大，预测值偏离真实值越厉害。偏差低意味着能较好地反应训练集中的数据情况。</p></li><li><p>方差(Variance)</p><p>指模型预测值的<strong>离散程度或者变化范围</strong>。方差越大，数据的分布越分散，函数波动越大，泛化能力越差。方差低意味着拟合曲线的稳定性高，波动小。</p></li></ul><p>据此，我们有对同一数据的各类拟合情况如下图：
-<img src='image/20180112_085630.png' alt='' /></p><p>据上图，高偏差意味着欠拟合，高方差意味着过拟合。</p><p>我们应尽量使得拟合模型处于低方差（较好地拟合数据）状态且同时处于低偏差（较好地预测新值）的状态。</p><p>避免过拟合的方法有：</p><ul><li><p>减少特征的数量</p><ul><li>手动选取需保留的特征</li><li>使用模型选择算法来选取合适的特征(如 PCA 算法)</li><li>减少特征的方式易丢失有用的特征信息</li></ul></li><li><p>正则化(Regularization)</p><ul><li>可保留所有参数（许多有用的特征都能轻微影响结果）</li><li>减少/惩罚各参数大小(magnitude)，以减轻各参数对模型的影响程度</li><li>当有很多参数对于模型只有轻微影响时，正则化方法的表现很好</li></ul></li></ul><h2><a name='header-n366' class='md-header-anchor '></a>7.2 代价函数(Cost Function)</h2><p>很多时候由于特征数量过多，过拟合时我们很难选出要保留的特征，这时候应用正则化方法则是很好的选择。</p><p>上文中，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-261-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="30.567ex" height="2.694ex" viewBox="0 -906.7 13160.5 1160" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E271-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 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xlink:href="#E274-MJMAIN-30" x="1501" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E274-MJMAIN-22C5" x="17270" y="0"></use><g transform="translate(17771,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E274-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E274-MJMAIN-32" x="663" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E274-MJMAIN-33" x="663" y="-434"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E274-MJMAIN-2B" x="18916" y="0"></use><g transform="translate(19917,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E274-MJMAIN-31"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E274-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E274-MJMAIN-30" x="1001" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E274-MJMAIN-30" x="1501" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E274-MJMAIN-22C5" x="22141" y="0"></use><g transform="translate(22642,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E274-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E274-MJMAIN-32" x="663" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E274-MJMAIN-34" x="663" y="-446"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-264">min_\theta\ \dfrac{1}{2m}\sum_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)})^2 + 1000\cdot\theta_3^2 + 1000\cdot\theta_4^2</script></p><p>上式中，我们在代价函数中增加了 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-268-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.414ex" viewBox="0 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332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E278-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E278-MJMAIN-33" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-268">\theta_3</script>、<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-269-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.414ex" viewBox="0 -790.8 923.4 1039.3" role="img" focusable="false" style="vertical-align: -0.577ex;"><defs><path stroke-width="1" id="E279-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E279-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E279-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E279-MJMAIN-34" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-269">\theta_4</script> 的惩罚项(penalty term) <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-267-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="19.788ex" height="3.044ex" viewBox="0 -906.7 8519.6 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E277-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E277-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 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149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E277-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E277-MJMAIN-30" x="1501" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E277-MJMAIN-22C5" x="2224" y="0"></use><g transform="translate(2724,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E277-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E277-MJMAIN-32" x="663" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E277-MJMAIN-33" x="663" y="-434"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E277-MJMAIN-2B" x="3870" y="0"></use><g transform="translate(4871,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E277-MJMAIN-31"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E277-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E277-MJMAIN-30" x="1001" y="0"></use><use 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366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E278-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E278-MJMAIN-33" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-268">\theta_3</script>、<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-269-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.414ex" viewBox="0 -790.8 923.4 1039.3" role="img" focusable="false" style="vertical-align: -0.577ex;"><defs><path stroke-width="1" id="E279-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E279-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E279-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E279-MJMAIN-34" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-269">\theta_4</script></strong>，从而使得假设函数中的 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-270-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.529ex" height="2.694ex" viewBox="0 -906.7 1949.8 1160" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E280-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E280-MJMAIN-33" d="M127 463Q100 463 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145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E280-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E280-MJMAIN-33" x="663" y="-213"></use><g transform="translate(923,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E280-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E280-MJMAIN-33" x="809" y="513"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-270">\theta_3x^3</script>、<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-271-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.529ex" height="2.577ex" viewBox="0 -906.7 1949.8 1109.7" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E281-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E281-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path stroke-width="1" id="E281-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" 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class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-277-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.355ex" height="1.994ex" viewBox="0 -755.9 583.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E287-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E287-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-277">\lambda</script>: 正则化参数(Regularization Parameter)，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-274-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.616ex" height="1.994ex" viewBox="0 -755.9 2418.1 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E284-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="1" id="E284-MJMAIN-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 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y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E286-MJMAIN-32" x="663" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E286-MJMATHI-6A" x="663" y="-430"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-276">\lambda \sum\limits_{j=1}^{n}{\theta_{j}^{2}}</script>: 正则化项</p></blockquote><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-277-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.355ex" height="1.994ex" viewBox="0 -755.9 583.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E287-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 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189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E294-MJMATHI-6E" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E294-MJMAIN-2B" x="822" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E294-MJMAIN-31" x="1823" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-284">n+1</script> 维单位矩阵</p></blockquote><p>Matlab/Octave 代码：</p><pre class="md-fences md-end-block md-fences-with-lineno" lang=""> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>10</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">&gt;&gt; L = eye(5)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">&gt;&gt; L(1,1) = 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">L =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 1 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 1 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 1 &nbsp; &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 1</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 224px;"></div><div class="CodeMirror-gutters" style="height: 254px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><p>&nbsp;</p><p>前文提到正则化可以解决正规方程法中不可逆的问题，即增加了 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-285-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.617ex" height="1.994ex" viewBox="0 -755.9 1987.9 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E295-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="1" id="E295-MJMAIN-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path stroke-width="1" id="E295-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E295-MJMATHI-3BB" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E295-MJMAIN-22C5" x="805" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E295-MJMATHI-4C" x="1306" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-285">\lambda \cdot L</script> 正则化项后，可以保证 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-286-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.824ex" height="2.461ex" viewBox="0 -906.7 5521.3 1059.4" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="1" id="E296-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E296-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E296-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="1" id="E296-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="1" id="E296-MJMAIN-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path stroke-width="1" id="E296-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E296-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E296-MJMATHI-54" x="1215" y="513"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E296-MJMATHI-58" x="1457" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E296-MJMAIN-2B" x="2532" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E296-MJMATHI-3BB" x="3533" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E296-MJMAIN-22C5" x="4339" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E296-MJMATHI-4C" x="4839" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-286">X^TX + \lambda \cdot L</script> 可逆(invertible)，即便 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-287-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.366ex" height="2.344ex" viewBox="0 -906.7 2310.4 1009.2" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E297-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E297-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-54" x="1215" y="513"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-58" x="1457" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-287">X^TX</script> 不可逆(non-invertible)。 </p><h2><a name='header-n453' class='md-header-anchor '></a>7.4 逻辑回归正则化(Regularized Logistic Regression)</h2><p>为逻辑回归的代价函数添加正则化项：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-288-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="89.471ex" height="4.095ex" viewBox="0 -1158 38522.3 1763.1" role="img" focusable="false" style="vertical-align: -1.405ex;"><defs><path stroke-width="1" id="E298-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E298-MJMAIN-28" d="M94 250Q94 319 104 381T127 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+<img src='image/20180112_085630.png' alt='' /></p><p>据上图，高偏差意味着欠拟合，高方差意味着过拟合。</p><p>我们应尽量使得拟合模型处于低方差（较好地拟合数据）状态且同时处于低偏差（较好地预测新值）的状态。</p><p>避免过拟合的方法有：</p><ul><li><p>减少特征的数量</p><ul><li>手动选取需保留的特征</li><li>使用模型选择算法来选取合适的特征(如 PCA 算法)</li><li>减少特征的方式易丢失有用的特征信息</li></ul></li><li><p>正则化(Regularization)</p><ul><li>可保留所有参数（许多有用的特征都能轻微影响结果）</li><li>减少/惩罚各参数大小(magnitude)，以减轻各参数对模型的影响程度</li><li>当有很多参数对于模型只有轻微影响时，正则化方法的表现很好</li></ul></li></ul><h2><a name='header-n365' class='md-header-anchor '></a>7.2 代价函数(Cost Function)</h2><p>很多时候由于特征数量过多，过拟合时我们很难选出要保留的特征，这时候应用正则化方法则是很好的选择。</p><p>上文中，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-166-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="30.567ex" height="2.694ex" viewBox="0 -906.7 13160.5 1160" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E168-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 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xlink:href="#E171-MJMAIN-30" x="1501" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E171-MJMAIN-22C5" x="17270" y="0"></use><g transform="translate(17771,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E171-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E171-MJMAIN-32" x="663" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E171-MJMAIN-33" x="663" y="-434"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E171-MJMAIN-2B" x="18916" y="0"></use><g transform="translate(19917,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E171-MJMAIN-31"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E171-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E171-MJMAIN-30" x="1001" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E171-MJMAIN-30" x="1501" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E171-MJMAIN-22C5" x="22141" y="0"></use><g transform="translate(22642,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E171-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E171-MJMAIN-32" x="663" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E171-MJMAIN-34" x="663" y="-446"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-169">min_\theta\ \dfrac{1}{2m}\sum_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)})^2 + 1000\cdot\theta_3^2 + 1000\cdot\theta_4^2</script></p><p>上式中，我们在代价函数中增加了 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-173-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.414ex" viewBox="0 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E176-MJMAIN-34" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-174">\theta_4</script> 的惩罚项(penalty term) <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-172-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="19.788ex" height="3.044ex" viewBox="0 -906.7 8519.6 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E174-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E174-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 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366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E175-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E175-MJMAIN-33" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-173">\theta_3</script>、<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-174-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.414ex" viewBox="0 -790.8 923.4 1039.3" role="img" focusable="false" style="vertical-align: -0.577ex;"><defs><path stroke-width="1" id="E176-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E176-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E176-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E176-MJMAIN-34" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-174">\theta_4</script></strong>，从而使得假设函数中的 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-175-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.529ex" height="2.694ex" viewBox="0 -906.7 1949.8 1160" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E177-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E177-MJMAIN-33" d="M127 463Q100 463 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145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E177-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E177-MJMAIN-33" x="663" y="-213"></use><g transform="translate(923,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E177-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E177-MJMAIN-33" x="809" y="513"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-175">\theta_3x^3</script>、<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-176-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.529ex" height="2.577ex" viewBox="0 -906.7 1949.8 1109.7" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E178-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E178-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path stroke-width="1" id="E178-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E178-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E178-MJMAIN-34" x="663" y="-213"></use><g transform="translate(923,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E178-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E178-MJMAIN-34" x="809" y="513"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-176">\theta_4x^4</script> 这两项的参数非常小，就相当于没有了，假设函数也就<strong>“变得”简单</strong>了，从而在保留各参数的情况下避免了过拟合问题。</p><p><img src='image/20180114_090054.png' alt='' /></p><p>&nbsp;</p><p>根据上面的讨论，有时也无法决定要减少哪个参数，故统一惩罚除了 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-182-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E184-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E184-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use 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y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E185-MJMAIN-32" x="663" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E185-MJMATHI-6A" x="663" y="-430"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-183">\lambda \sum\limits_{j=1}^{n}{\theta_{j}^{2}}</script>: 正则化项</p></blockquote><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-184-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.355ex" height="1.994ex" viewBox="0 -755.9 583.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E186-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E186-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-184">\lambda</script> 正则化参数类似于学习速率，也需要我们自行对其选择一个合适的值。</p><ul><li><p>过大</p><ul><li>导致模型欠拟合(假设可能会变成近乎 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-185-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.573ex" height="2.461ex" viewBox="0 -806.1 2830 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E187-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 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y="584"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E190-MJMATHI-6D" x="84" y="-488"></use></g><g transform="translate(981,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E190-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E190-MJMATHI-6A" x="663" y="-213"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-188">\frac{\lambda}{m}\theta_j</script>: 正则化项</p></blockquote><p>&nbsp;</p><p>应用正则化的正规方程法<sup><a href='#dfref-footnote-2' name='ref-footnote-2'>2</a></sup>：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-189-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="33.291ex" height="21.488ex" viewBox="0 -4877.1 14333.7 9251.7" role="img" focusable="false" style="vertical-align: -9.944ex; margin-bottom: -0.217ex;"><defs><path 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-2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="1" id="E192-MJMAIN-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path stroke-width="1" id="E192-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g 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656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E3-MJMATHI-4C" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-3">L</script>: 第一行第一列为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-191-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.162ex" height="1.994ex" viewBox="0 -755.9 500.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E193-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E193-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-191">0</script> 的 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-192-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.398ex" height="2.11ex" viewBox="0 -755.9 2323.9 908.7" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="1" id="E194-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E194-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="1" id="E194-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E194-MJMATHI-6E" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E194-MJMAIN-2B" x="822" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E194-MJMAIN-31" x="1823" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-192">n+1</script> 维单位矩阵</p></blockquote><p>Matlab/Octave 代码：</p><pre class="md-fences md-end-block md-fences-with-lineno" lang=""> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>10</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">&gt;&gt; L = eye(5)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">&gt;&gt; L(1,1) = 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">L =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 1 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 1 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 1 &nbsp; &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 1</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 224px;"></div><div class="CodeMirror-gutters" style="height: 254px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><p>&nbsp;</p><p>前文提到正则化可以解决正规方程法中不可逆的问题，即增加了 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-193-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.617ex" height="1.994ex" viewBox="0 -755.9 1987.9 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E195-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="1" id="E195-MJMAIN-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path stroke-width="1" id="E195-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMATHI-3BB" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMAIN-22C5" x="805" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E195-MJMATHI-4C" x="1306" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-193">\lambda \cdot L</script> 正则化项后，可以保证 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-194-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.824ex" height="2.461ex" viewBox="0 -906.7 5521.3 1059.4" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="1" id="E196-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E196-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E196-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="1" id="E196-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="1" id="E196-MJMAIN-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path stroke-width="1" id="E196-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E196-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E196-MJMATHI-54" x="1215" y="513"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E196-MJMATHI-58" x="1457" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E196-MJMAIN-2B" x="2532" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E196-MJMATHI-3BB" x="3533" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E196-MJMAIN-22C5" x="4339" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E196-MJMATHI-4C" x="4839" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-194">X^TX + \lambda \cdot L</script> 可逆(invertible)，即便 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-195-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.366ex" height="2.344ex" viewBox="0 -906.7 2310.4 1009.2" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E197-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E197-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 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xlink:href="#E200-MJSZ4-5D" x="16710" y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-2C" x="24075" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMATHI-6A" x="25270" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-2208" x="25960" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-7B" x="26906" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-31" x="27406" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-2C" x="27907" y="0"></use><g transform="translate(28352,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-32"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-2E" x="779" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-2E" x="1057" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMATHI-6E" x="29688" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-7D" x="30288" y="0"></use></g><g transform="translate(0,-3883)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E200-MJMAIN-7D" x="0" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-198">\begin{align*} & \text{Repeat}\ \lbrace \newline & \ \ \ \ \theta_0 := \theta_0 - \alpha\ \frac{1}{m}\ \sum_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)})x_0^{(i)} \newline & \ \ \ \ \theta_j := \theta_j - \alpha\ \left[ \left( \frac{1}{m}\ \sum_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)})x_j^{(i)} \right) + \frac{\lambda}{m}\theta_j \right], \ \ \ j \in \lbrace 1,2...n\rbrace\newline & \rbrace \end{align*}</script></p><p>&nbsp;</p><div class='footnotes-area'  ><hr/>
 <div class='footnote-line'><span class='md-fn-count'>1</span> <a href='https://en.wikipedia.org/wiki/List_of_algorithms#Optimization_algorithms' target='_blank' >https://en.wikipedia.org/wiki/List_of_algorithms#Optimization_algorithms</a><a name='dfref-footnote-1' href='#ref-footnote-1' title='回到文档' class='reversefootnote' >↩</a></div>
 <div class='footnote-line'><span class='md-fn-count'>2</span> week2 - 4.6<a name='dfref-footnote-2' href='#ref-footnote-2' title='回到文档' class='reversefootnote' >↩</a></div></div></div>
 </body>
diff --git a/week3.md b/week3.md
index 27b368f..cef75bf 100644
--- a/week3.md
+++ b/week3.md
@@ -173,29 +173,28 @@ $J(\theta) = - \frac{1}{m} \displaystyle \sum_{i=1}^m [y^{(i)}\log (h_\theta (x^
 
 令 $f(\theta) = {{y}^{(i)}}\log \left( {h_\theta}\left( {{x}^{(i)}} \right) \right)+\left( 1-{{y}^{(i)}} \right)\log \left( 1-{h_\theta}\left( {{x}^{(i)}} \right) \right)$
 
-将 $h_\theta(x^{(i)}) = g\left(\theta^{T}x^{(i)} \right)=\frac{1}{1+{{e}^{-{\theta^T}{{x}^{(i)}}}}} $ 带入得
+忆及 $z=\Theta^Tx^{(i)}$，将 $h_\theta(x^{(i)}) = g(z) =\frac{1}{1+{{e}^{-{\theta^T}{{x}^{(i)}}}}} $ 带入得
 
 $f(\theta)={{y}^{(i)}}\log \left( \frac{1}{1+{{e}^{-{\theta^T}{{x}^{(i)}}}}} \right)+\left( 1-{{y}^{(i)}} \right)\log \left( 1-\frac{1}{1+{{e}^{-{\theta^T}{{x}^{(i)}}}}} \right)$
-$=-{{y}^{(i)}}\log \left( 1+{{e}^{-{\theta^T}{{x}^{(i)}}}} \right)-\left( 1-{{y}^{(i)}} \right)\log \left( 1+{{e}^{{\theta^T}{{x}^{(i)}}}} \right)$
+$=-{{y}^{(i)}}\log \left( 1+{{e}^{-z}} \right)-\left( 1-{{y}^{(i)}} \right)\log \left( 1+{{e}^{z}} \right)$
 
 根据求偏导的性质，没有 $\theta_j$ 的项求偏导即为 $0$，都消去，则得：
 
-$\frac{\partial }{\partial {\theta_{j}}}\left( \theta^Tx^{(i)}  \right)=x^{(i)}_j$
+$\frac{\partial z}{\partial {\theta_{j}}}=\frac{\partial }{\partial {\theta_{j}}}\left( \theta^Tx^{(i)}  \right)=x^{(i)}_j$
 
 所以有：
 
-$\frac{\partial }{\partial {\theta_{j}}}f\left( \theta  \right)=\frac{\partial }{\partial {\theta_{j}}}[-{{y}^{(i)}}\log \left( 1+{{e}^{-{\theta^{T}}{{x}^{(i)}}}} \right)-\left( 1-{{y}^{(i)}} \right)\log \left( 1+{{e}^{{\theta^{T}}{{x}^{(i)}}}} \right)]$
+$\frac{\partial }{\partial {\theta_{j}}}f\left( \theta  \right)=\frac{\partial }{\partial {\theta_{j}}}[-{{y}^{(i)}}\log \left( 1+{{e}^{-z}} \right)-\left( 1-{{y}^{(i)}} \right)\log \left( 1+{{e}^{z}} \right)]$
 
-$=-{{y}^{(i)}}\frac{\frac{\partial }{\partial {\theta_{j}}}\left( \theta^Tx^{(i)}  \right)\cdot{{e}^{-{\theta^{T}}{{x}^{(i)}}}}}{1+{{e}^{-{\theta^{T}}{{x}^{(i)}}}}}-\left( 1-{{y}^{(i)}} \right)\frac{\frac{\partial }{\partial {\theta_{j}}}\left( \theta^Tx^{(i)}  \right)\cdot{{e}^{{\theta^T}{{x}^{(i)}}}}}{1+{{e}^{{\theta^T}{{x}^{(i)}}}}}$
+$=-{{y}^{(i)}}\frac{\frac{\partial }{\partial {\theta_{j}}}\left(-z \right) e^{-z}}{1+e^{-z}}-\left( 1-{{y}^{(i)}} \right)\frac{\frac{\partial }{\partial {\theta_{j}}}\left(z \right){e^{z}}}{1+e^{z}}$
 
-$=-{{y}^{(i)}}\frac{-x_{j}^{(i)}{{e}^{-{\theta^{T}}{{x}^{(i)}}}}}{1+{{e}^{-{\theta^{T}}{{x}^{(i)}}}}}-\left( 1-{{y}^{(i)}} \right)\frac{x_j^{(i)}{{e}^{{\theta^T}{{x}^{(i)}}}}}{1+{{e}^{{\theta^T}{{x}^{(i)}}}}}$
-$={{y}^{(i)}}\frac{x_j^{(i)}}{1+{{e}^{{\theta^T}{{x}^{(i)}}}}}-\left( 1-{{y}^{(i)}} \right)\frac{x_j^{(i)}{{e}^{{\theta^T}{{x}^{(i)}}}}}{1+{{e}^{{\theta^T}{{x}^{(i)}}}}}$ 
-$={\frac{{{y}^{(i)}}x_j^{(i)}-x_j^{(i)}{{e}^{{\theta^T}{{x}^{(i)}}}}+{{y}^{(i)}}x_j^{(i)}{{e}^{{\theta^T}{{x}^{(i)}}}}}{1+{{e}^{{\theta^T}{{x}^{(i)}}}}}}$
-$={\frac{{{y}^{(i)}}\left( 1\text{+}{{e}^{{\theta^T}{{x}^{(i)}}}} \right)-{{e}^{{\theta^T}{{x}^{(i)}}}}}{1+{{e}^{{\theta^T}{{x}^{(i)}}}}}x_j^{(i)}}$
-$={({{y}^{(i)}}-\frac{{{e}^{{\theta^T}{{x}^{(i)}}}}}{1+{{e}^{{\theta^T}{{x}^{(i)}}}}})x_j^{(i)}}$
-$={({{y}^{(i)}}-\frac{1}{1+{{e}^{-{\theta^T}{{x}^{(i)}}}}})x_j^{(i)}}$
+$=-{{y}^{(i)}}\frac{-x^{(i)}_je^{-z}}{1+e^{-z}}-\left( 1-{{y}^{(i)}} \right)\frac{x^{(i)}_j}{1+e^{-z}}$
+$=\left({{y}^{(i)}}\frac{e^{-z}}{1+e^{-z}}-\left( 1-{{y}^{(i)}} \right)\frac{1}{1+e^{-z}}\right)x^{(i)}_j$ 
+$=\left({{y}^{(i)}}\frac{e^{-z}}{1+e^{-z}}-\left( 1-{{y}^{(i)}} \right)\frac{1}{1+e^{-z}}\right)x^{(i)}_j$
+$=\left(\frac{{{y}^{(i)}}(e^{-z}+1)-1}{1+e^{-z}}\right)x^{(i)}_j$
+$={({{y}^{(i)}}-\frac{1}{1+{{e}^{-z}}})x_j^{(i)}}$
 $={\left({{y}^{(i)}}-{h_\theta}\left( {{x}^{(i)}} \right)\right)x_j^{(i)}}$
-$={\left({h_\theta}\left( {{x}^{(i)}} \right)-{{y}^{(i)}}\right)x_j^{(i)}}$
+$=-{\left({h_\theta}\left( {{x}^{(i)}} \right)-{{y}^{(i)}}\right)x_j^{(i)}}$
 
 则可得代价函数的导数：
 
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-<div  id='write'  class = 'is-node show-fences-line-number'><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n2">8 神经网络：表达(Neural Networks: Representation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n3">8.1 非线性假设(Non-linear Hypotheses)</a></span><span class="md-toc-item md-toc-h2" data-ref="n14"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n14">8.2 神经网络和大脑(Neurons and the Brain)</a></span><span class="md-toc-item md-toc-h2" data-ref="n29"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n29">8.3 模型表示1(Model Representation I)</a></span><span class="md-toc-item md-toc-h2" data-ref="n96"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n96">8.4 模型表示2(Model Representation II)</a></span><span class="md-toc-item md-toc-h2" data-ref="n150"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n150">8.5 例子和直观理解1(Examples and Intuitions I)</a></span><span class="md-toc-item md-toc-h2" data-ref="n173"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n173">8.6 例子和直观理解2(Examples and Intuitions II)</a></span><span class="md-toc-item md-toc-h2" data-ref="n195"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n195">8.7 多类别分类(Multiclass Classification)</a></span></p></div><h1><a name='header-n2' class='md-header-anchor '></a>8 神经网络：表达(Neural Networks: Representation)</h1><h2><a name='header-n3' class='md-header-anchor '></a>8.1 非线性假设(Non-linear Hypotheses)</h2><p>理论上我们可以用多项式函数去近似任意函数（泰勒极数(Taylor series)），从而可得到任意问题的拟合曲线。</p><p>在实际处理时，特征量通常会很多，如果再构造高阶多项式等，特征数量将会急剧增加，这使得回归模型的复杂度太高，可见并不合适。神经网络无需构造高阶多项式，在特征量很大时也可以处理的很好。</p><p>那特征能有多大呢？下面是一个计算机视觉中的例子：</p><p><img src='image/20180115_084326.png' alt='' /></p><p>如上图，如果选取一小块 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-86-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.845ex" height="1.994ex" viewBox="0 -755.9 2946.9 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E86-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path stroke-width="1" id="E86-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E86-MJMAIN-2217" d="M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E86-MJMAIN-35"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E86-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E86-MJMAIN-2217" x="1223" y="0"></use><g transform="translate(1945,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E86-MJMAIN-35"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E86-MJMAIN-30" x="500" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-86">50 * 50</script> 像素的灰度图片（一个像素只有亮度一个值），选择每个像素点作为特征，则特征总量 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-87-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.143ex" height="1.994ex" viewBox="0 -755.9 3936.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E87-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E87-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E87-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E87-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path stroke-width="1" id="E87-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMATHI-6E" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-3D" x="878" y="0"></use><g transform="translate(1934,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-32"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-35" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-30" x="1001" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E87-MJMAIN-30" x="1501" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-87">n=2500</script>（换成 RGB（一个像素有三个值），则 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-88-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.143ex" height="1.994ex" viewBox="0 -755.9 3936.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E88-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E88-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E88-MJMAIN-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path><path stroke-width="1" id="E88-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path stroke-width="1" id="E88-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E88-MJMATHI-6E" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E88-MJMAIN-3D" x="878" y="0"></use><g transform="translate(1934,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E88-MJMAIN-37"></use><use 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413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path><path stroke-width="1" id="E89-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E89-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path stroke-width="1" id="E89-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E89-MJMAIN-2248" d="M55 319Q55 360 72 393T114 444T163 472T205 482Q207 482 213 482T223 483Q262 483 296 468T393 413L443 381Q502 346 553 346Q609 346 649 375T694 454Q694 465 698 474T708 483Q722 483 722 452Q722 386 675 338T555 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xlink:href="#E89-MJMAIN-33" x="3565" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E89-MJMATHI-6D" x="4315" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E89-MJMATHI-69" x="5194" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E89-MJMATHI-6C" x="5539" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E89-MJMATHI-6C" x="5838" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E89-MJMATHI-69" x="6136" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E89-MJMATHI-6F" x="6482" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E89-MJMATHI-6E" x="6967" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-89">C_{2500}^{2} \approx 3\ million</script>。</p><h2><a name='header-n14' class='md-header-anchor '></a>8.2 神经网络和大脑(Neurons and the Brain)</h2><p>脑科学家通过对动物实验，发现大脑中专用于处理听觉信号的脑皮层也能处理其他诸如视觉等信号，即如果切断其与耳朵的联系，将其与眼睛相连，则这块负责听觉的脑皮层区域也能接受并处理视觉信号，从而学会“看”。脑科学家通过这类换源实验，就推论假设大脑的学习算法只有一种(“one learning algorithm” hypothesis)。那么如果能找出这种学习算法并应用于计算机中，那梦想中和人一样的人工智能就成真了。</p><p>神经网络就源于<strong>模拟人类大脑</strong>，但其需要的计算量很大。随着计算机硬件性能的提高，神经网络逐渐从衰落变为流行，如今已广泛地被应用在各行各业中。</p><p>下图是根据研究做的一些应用（有兴趣可回顾视频）：</p><p><img src='image/20180115_101441.png' alt='' /></p><p>BrainPort  系统：帮助失明人士通过摄像头以及舌尖感官“看”东西</p><p><img src='image/20180115_101442.png' alt='' /></p><p>触觉皮带：在朝北时蜂鸣器会发出声响，可使人拥有方向感（声音信号转换为方向信号）。</p><h2><a name='header-n29' class='md-header-anchor '></a>8.3 模型表示1(Model Representation I)</h2><p>既然神经网络模仿的是大脑神经元，那就先看一下大脑的神经元长什么样吧：</p><p><img src='image/20141213201613758.jpg' alt='来源: http://blog.csdn.net/zzwu/article/details/574931' /></p><p>想象一下印刷厂中流水线的工人（机器人也算哦），每个工人都有特定的任务，比如装订，塑封，贴防伪标识等等，工人们看到书本并处理完自己的任务后，就回放回传送带，紧接着传送带就传给下一个环节的工人，如此不断重复从而完成一个又一个环节，直到一本书印制完成。</p><p>那么类比一下，把上图中的<strong>细胞核(nucleus)</strong>类比成工人，<strong>轴突(axon)</strong>类比传送带，<strong>树突(dendrite)</strong>则比类比成工人的双眼。一个又一个细胞体，从树突接收需要处理的信息，对其进行处理后，再经由轴突通过电信号把处理完的信息传递出去，直到理解信息的内容。当然啦，我们大脑的实际上还要更为复杂，而且一个人的神经元数目就比地球上所有流水线的工人之和还要多呢~</p><p>人工神经网络中，树突对应<strong>输入(input)</strong>，细胞核对应<strong>激活单元(activation unit)</strong>，轴突对应<strong>输出(output)</strong>。</p><p>我们一般把神经网络划分为三部分（注意，不是只有三层！），即输入层(input layer)，隐藏层(hidden layer)和输出层(output layer)。</p><p><img src='image/20180116_001543.png' alt='' /></p><p>图中的一个圈表示神经网络中的一个激活单元，输入层对应输入单元，隐藏层对应中间单元，输出层则对应输出单元。中间激活单元应用<strong>激活函数</strong>(<a href='https://en.wikipedia.org/wiki/Activation_function'>activation_function</a>)处理数据。</p><p>下面列出一些已有概念在神经网络中的别称：</p><ul><li><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-91-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.384ex" height="1.76ex" viewBox="0 -504.6 1026.4 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E91-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E91-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E91-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E91-MJMAIN-30" x="809" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-91">x_0</script>: 偏置单元(bias unit)，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-91-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.384ex" height="1.76ex" viewBox="0 -504.6 1026.4 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E91-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E91-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E91-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E91-MJMAIN-30" x="809" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-91">x_0</script>=1</li><li><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-306-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.808ex" height="2.11ex" viewBox="0 -806.1 778.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E322-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E322-MJMAIN-398" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-306">\Theta</script>: 权重(weight)，即参数。</li><li>激活函数: <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-92-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.116ex" height="1.877ex" viewBox="0 -504.6 480.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" 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width="1.128ex" height="1.41ex" viewBox="0 -504.6 485.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E99-MJMATHI-76" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E99-MJMATHI-76" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-99">v</script> 个单元的权重</p><p><span class="MathJax_Preview"></span><span 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E105-MJMATHI-6A" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E105-MJMAIN-2B" x="4461" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E105-MJMAIN-31" x="5462" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E105-MJMAIN-29" x="5962" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-105">s_{j+1} \times (s_j+1)</script> 维的权重矩阵。即每一层的权重矩阵大小都是非固定的。</li><li>其中，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-106-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.971ex" height="2.11ex" viewBox="0 -755.9 1279 908.7" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="1" id="E106-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-31" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-2B" x="17189" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-31" x="18190" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-29" x="18691" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-3D" x="19358" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-33" x="20414" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-D7" x="21137" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-34" x="22138" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-107">Size(\Theta^{(1)})=s_{j+1} \times (s_j + 1) =s_2 \times (s_1 + 1) = 3 \times 4</script></p><p><span 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xlink:href="#E111-MJMAIN-29" x="18645" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-111">\begin{align*} a_1^{(2)} = g(\Theta_{10}^{(1)}x_0 + \Theta_{11}^{(1)}x_1 + \Theta_{12}^{(1)}x_2 + \Theta_{13}^{(1)}x_3) \newline a_2^{(2)} = g(\Theta_{20}^{(1)}x_0 + \Theta_{21}^{(1)}x_1 + \Theta_{22}^{(1)}x_2 + \Theta_{23}^{(1)}x_3) \newline a_3^{(2)} = g(\Theta_{30}^{(1)}x_0 + \Theta_{31}^{(1)}x_1 + \Theta_{32}^{(1)}x_2 + \Theta_{33}^{(1)}x_3) \newline \end{align*}</script></p><p>对第 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-112-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.162ex" height="1.994ex" viewBox="0 -755.9 500.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E112-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 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type="math/tex" id="MathJax-Element-113">h_\Theta(x) = a_1^{(3)} = g(\Theta_{10}^{(2)}a_0^{(2)} + \Theta_{11}^{(2)}a_1^{(2)} + \Theta_{12}^{(2)}a_2^{(2)} + \Theta_{13}^{(2)}a_3^{(2)})</script></p><p>上面的计算过程被称为<strong>前向传播(Forward propagation)</strong>，即从输入层开始，一层一层地向下计算并传递结果。</p><p>&nbsp;</p><p>再回顾一下逻辑回归：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-114-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="36.531ex" height="2.577ex" viewBox="0 -806.1 15728.6 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E114-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 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x="0" y="-4285"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-117">\Theta^{(1)}=\left[\begin{matrix}\Theta^{(1)}_{10}& \Theta^{(1)}_{11}& \Theta^{(1)}_{12}& \Theta^{(1)}_{13}\\ \Theta^{(1)}_{20}& \Theta^{(1)}_{21}& \Theta^{(1)}_{22}& \Theta^{(1)}_{23}\\ \Theta^{(1)}_{30}& \Theta^{(1)}_{31}& \Theta^{(1)}_{32} & \Theta^{(1)}_{33}\end{matrix}\right]</script>，</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-118-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.752ex" height="11.216ex" viewBox="0 -2665.7 5921.1 4828.9" role="img" focusable="false" style="vertical-align: -4.874ex; margin-bottom: -0.15ex;"><defs><path stroke-width="1" id="E118-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 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xmlns:xlink="http://www.w3.org/1999/xlink" width="26.74ex" height="2.928ex" viewBox="0 -956.9 11513 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E120-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="1" id="E120-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 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168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="1" id="E120-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E120-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 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xlink:href="#E120-MJMAIN-28" x="3348" y="0"></use><g transform="translate(3738,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-398" x="0" y="0"></use><g transform="translate(778,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-31" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-29" x="890" y="0"></use></g></g><g transform="translate(5521,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-31" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-29" x="890" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-29" x="7055" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-3D" x="7723" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMATHI-67" x="8779" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-28" x="9259" y="0"></use><g transform="translate(9649,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMATHI-7A" x="0" y="0"></use><g transform="translate(469,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E120-MJMAIN-32" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" 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xlink:href="#E122-MJMATHI-6E" x="748" y="-212"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-122">z^{(j)}_i = \Theta^{(j-1)}_{i,0}a^{(j-1)}_{0}+ \Theta^{(j-1)}_{i,1}a^{(j-1)}_{1}+\dots+ \Theta^{(j-1)}_{i,n}a^{(j-1)}_{n}</script>，</p><p> <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-123-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.995ex" height="2.461ex" viewBox="0 -956.9 7747.8 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E123-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E123-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E123-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E123-MJMATHI-6A" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E123-MJMAIN-2212" x="802" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E123-MJMAIN-31" x="1580" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E123-MJMAIN-29" x="2081" y="0"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-123">z^{(j)} = \Theta^{(j-1)}a^{(j-1)}</script>，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-124-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.722ex" height="2.928ex" viewBox="0 -956.9 5477.5 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E124-MJMATHI-61" d="M33 157Q33 258 109 349T280 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71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E124-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E124-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E124-MJMATHI-6A" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E124-MJMAIN-29" x="802" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E124-MJMAIN-3D" x="1749" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E124-MJMATHI-67" x="2806" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E124-MJMAIN-28" x="3286" y="0"></use><g transform="translate(3676,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E124-MJMATHI-7A" x="0" y="0"></use><g transform="translate(469,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E124-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E124-MJMATHI-6A" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E124-MJMAIN-29" x="802" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E124-MJMAIN-29" x="5088" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-124">a^{(j)} = g(z^{(j)})</script>，通过该式即可计算神经网络中每一层的值。</p><p>扩展到所有样本实例：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" 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0Z"></path><path stroke-width="1" id="E126-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E126-MJMATHI-7A" x="0" y="0"></use><g transform="translate(469,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E126-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E126-MJMAIN-32" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E126-MJMAIN-29" x="890" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E126-MJMAIN-3D" x="1751" y="0"></use><g transform="translate(2808,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E126-MJMAIN-398" x="0" y="0"></use><g transform="translate(778,432)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E126-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E126-MJMAIN-31" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E126-MJMAIN-29" x="890" y="0"></use></g></g><g transform="translate(4591,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E126-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E126-MJMATHI-54" x="1215" y="581"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-126">{{z}^{\left( 2 \right)}}={{\Theta }^{\left( 1 \right)}} {{X}^{T}}</script>，这时 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-127-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.424ex" height="2.461ex" viewBox="0 -956.9 1474.2 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E127-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E127-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E127-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E127-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E127-MJMATHI-7A" x="0" y="0"></use><g transform="translate(469,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E127-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E127-MJMAIN-32" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E127-MJMAIN-29" x="890" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-127">z^{(2)}</script> 是一个 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-128-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.025ex" height="1.76ex" viewBox="0 -554.9 3024.9 757.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E128-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="1" id="E128-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E128-MJMAIN-D7" d="M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path><path stroke-width="1" id="E128-MJMATHI-6D" d="M21 287Q22 293 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xlink:href="#E128-MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E128-MJMAIN-32" x="663" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E128-MJMAIN-D7" x="1145" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E128-MJMATHI-6D" x="2146" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-128">s_2 \times m</script> 维矩阵。</p><blockquote><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-18-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.04ex" height="1.41ex" viewBox="0 -504.6 878.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E18-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E18-MJMATHI-6D" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-18">m</script>: 训练集中的样本实例数量</p><p><span 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class='md-header-anchor '></a>8.5 例子和直观理解1(Examples and Intuitions I)</h2><p>为了更好的理解神经网络，举例单层神经网络进行逻辑运算的例子。</p><p>下面的例子中，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-130-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.802ex" height="1.76ex" viewBox="0 -504.6 2498 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E130-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 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xlink:href="#E132-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E132-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E132-MJMAIN-29" x="13062" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-132">h_\Theta(x) = g(-30+20x_1+20x_2)</script>。</p><p>回顾 sigmoid 函数图像，根据输入则有上图中右边的表格，即 <span class="MathJax_Preview"></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-133-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-133">h_\theta(x)\approx x_1\ \text{AND}\ x_2</script>。这样就实现了一个能够进行与运算的神经网络。 </p><p><img src='image/2413fbec8ff9fa1f19aaf78265b8a33b_Logistic_function.png' alt='sigmoid function' /></p><p>&nbsp;</p><p>再举一例，逻辑或(OR)运算（有一个真值则结果就为真）神经网络：</p><p><img src='image/20180117_000349.png' alt='' /></p><p>&nbsp;</p><h2><a name='header-n173' class='md-header-anchor '></a>8.6 例子和直观理解2(Examples and Intuitions II)</h2><p>下面逐步构建复杂一点的神经网络</p><p><img src='image/20180117_004820.png' alt='' /></p><p>如上图，我们分别构建了三个单层神经网络，将这三个网络组合起来，可得到一个新的神经网络，其可完成逻辑运算中的异或(XNOR)操作：</p><p><img src='image/20180116_235545.png' alt='' /></p><p>这里的组合即为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-134-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="54.065ex" height="2.577ex" viewBox="0 -806.1 23278 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E133-MJMAIN-58" d="M270 0Q252 3 141 3Q46 3 31 0H23V46H40Q129 50 161 88Q165 94 244 216T324 339Q324 341 235 480T143 622Q133 631 119 634T57 637H37V683H46Q64 680 172 680Q297 680 318 683H329V637H324Q307 637 286 632T263 621Q263 618 322 525T384 431Q385 431 437 511T489 593Q490 595 490 599Q490 611 477 622T436 637H428V683H437Q455 680 566 680Q661 680 676 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a^{(2)}) \newline& h_\Theta(x) = a^{(3)}\end{align*}</script></p><p>可见，特征值能不断升级，并抽取出更多信息，直到计算出结果。而如此不断组合，我们就可以逐渐构造出越来越复杂、强大的神经网络，比如用于手写识别的神经网络。</p><blockquote><p>任何布尔函数都可由两层神经网络准确表达，但所需的中间单元的数量随输入呈指数级增长;</p><p>任何连续函数都可由两层神经网络以任意精度逼近;</p><p>任何函数都可由三层神经网络以任意程度逼近。</p></blockquote><h2><a name='header-n195' class='md-header-anchor '></a>8.7 多类别分类(Multiclass Classification)</h2><p>之前讨论的都是预测结果为单值情况下的神经网络，要实现多类别分类，其实只要修改一下输出层，让输出层包含多个输出单元即可。</p><p>举一个 4 分类问题的实例：</p><p><img src='image/20180117_010904.png' alt='' /></p><p>有四种分类情况，那么就让输出层包含 4 个输出单元即可，则 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-138-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.85ex" height="2.344ex" viewBox="0 -755.9 1227 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E137-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 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52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E138-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path stroke-width="1" id="E138-MJSZ4-23A1" d="M319 -645V1154H666V1070H403V-645H319Z"></path><path stroke-width="1" id="E138-MJSZ4-23A3" d="M319 -644V1155H403V-560H666V-644H319Z"></path><path stroke-width="1" id="E138-MJSZ4-23A2" d="M319 0V602H403V0H319Z"></path><path stroke-width="1" id="E138-MJSZ4-23A4" d="M0 1070V1154H347V-645H263V1070H0Z"></path><path stroke-width="1" id="E138-MJSZ4-23A6" d="M263 -560V1155H347V-644H0V-560H263Z"></path><path stroke-width="1" id="E138-MJSZ4-23A5" d="M263 0V602H347V0H263Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJMAIN-398" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJMAIN-28" x="1226" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJMATHI-78" x="1616" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJMAIN-29" x="2188" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJMAIN-3D" x="2856" y="0"></use><g transform="translate(3912,0)"><g transform="translate(0,2850)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJSZ4-23A1" x="0" y="-1155"></use><g transform="translate(0,-3446.188741721854) scale(1,2.8112582781456954)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJSZ4-23A2"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJSZ4-23A3" x="0" y="-4555"></use></g><g transform="translate(834,0)"><g transform="translate(-15,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJMAIN-30" x="0" y="2050"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJMAIN-30" x="0" y="650"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJMAIN-31" x="0" y="-750"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJMAIN-30" x="0" y="-2150"></use></g></g><g transform="translate(1487,2850)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJSZ4-23A4" x="0" y="-1155"></use><g transform="translate(0,-3446.188741721854) scale(1,2.8112582781456954)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJSZ4-23A5"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E138-MJSZ4-23A6" x="0" y="-4555"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-139">h_\Theta(x) =\begin{bmatrix}0 \newline 0 \newline 1 \newline 0 \newline\end{bmatrix}</script>，</p><p>那么表示 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-140-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.043ex" height="2.577ex" viewBox="0 -806.1 3032.4 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E139-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E139-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="1" id="E139-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E139-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E139-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E139-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E139-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E139-MJMAIN-398" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E139-MJMAIN-28" x="1226" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E139-MJMATHI-78" x="1616" y="0"></use><g transform="translate(2188,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E139-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E139-MJMAIN-33" x="550" y="-213"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-140">h_\Theta(x)_3</script>，即分为第 3 类，对应于图中的摩托车(Motorcycle)。</p><p><strong>总结一下</strong>，要分为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.211ex" height="1.994ex" viewBox="0 -755.9 521.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E143-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E143-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">k</script> 类，就在输出层放置 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.211ex" height="1.994ex" viewBox="0 -755.9 521.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E143-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E143-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">k</script> 个输出单元，对于单个样本实例，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-143-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.989ex" height="2.577ex" viewBox="0 -806.1 2578.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E142-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E142-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="1" id="E142-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E142-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E142-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 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id="E143-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E143-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-144">k</script> 维结果向量，最后依据结果向量，得出属于哪个类 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-145-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.24ex" height="2.928ex" viewBox="0 -956.9 1394.8 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E144-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 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380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E144-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E144-MJMATHI-79" x="0" y="0"></use><g transform="translate(499,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E144-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E144-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E144-MJMAIN-29" x="734" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-145">y^{(i)}</script>。</p><p>&nbsp;</p></div>
+<div  id='write'  class = 'is-node show-fences-line-number'><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" href="#header-n2">8 神经网络：表达(Neural Networks: Representation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" href="#header-n3">8.1 非线性假设(Non-linear Hypotheses)</a></span><span class="md-toc-item md-toc-h2" data-ref="n14"><a class="md-toc-inner" href="#header-n14">8.2 神经网络和大脑(Neurons and the Brain)</a></span><span class="md-toc-item md-toc-h2" data-ref="n29"><a class="md-toc-inner" href="#header-n29">8.3 模型表示1(Model Representation I)</a></span><span class="md-toc-item md-toc-h2" data-ref="n96"><a class="md-toc-inner" href="#header-n96">8.4 模型表示2(Model Representation II)</a></span><span class="md-toc-item md-toc-h2" data-ref="n150"><a class="md-toc-inner" href="#header-n150">8.5 例子和直观理解1(Examples and Intuitions I)</a></span><span class="md-toc-item md-toc-h2" data-ref="n173"><a class="md-toc-inner" href="#header-n173">8.6 例子和直观理解2(Examples and Intuitions II)</a></span><span class="md-toc-item md-toc-h2" data-ref="n195"><a class="md-toc-inner" href="#header-n195">8.7 多类别分类(Multiclass Classification)</a></span></p></div><h1><a name='header-n2' class='md-header-anchor '></a>8 神经网络：表达(Neural Networks: Representation)</h1><h2><a name='header-n3' class='md-header-anchor '></a>8.1 非线性假设(Non-linear Hypotheses)</h2><p>理论上我们可以用多项式函数去近似任意函数（泰勒极数(Taylor series)），从而可得到任意问题的拟合曲线。</p><p>在实际处理时，特征量通常会很多，如果再构造高阶多项式等，特征数量将会急剧增加，这使得回归模型的复杂度太高，可见并不合适。神经网络无需构造高阶多项式，在特征量很大时也可以处理的很好。</p><p>那特征能有多大呢？下面是一个计算机视觉中的例子：</p><p><img src='image/20180115_084326.png' alt='' /></p><p>如上图，如果选取一小块 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-204-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.845ex" height="1.994ex" viewBox="0 -755.9 2946.9 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E207-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path stroke-width="1" id="E207-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E207-MJMAIN-2217" d="M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-35"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-2217" x="1223" y="0"></use><g transform="translate(1945,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-35"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E207-MJMAIN-30" x="500" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-204">50 * 50</script> 像素的灰度图片（一个像素只有亮度一个值），选择每个像素点作为特征，则特征总量 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-205-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.143ex" height="1.994ex" viewBox="0 -755.9 3936.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E208-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E208-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E208-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E208-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path stroke-width="1" id="E208-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E208-MJMATHI-6E" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E208-MJMAIN-3D" x="878" y="0"></use><g transform="translate(1934,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E208-MJMAIN-32"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E208-MJMAIN-35" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E208-MJMAIN-30" x="1001" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E208-MJMAIN-30" x="1501" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-205">n=2500</script>（换成 RGB（一个像素有三个值），则 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-206-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.143ex" height="1.994ex" viewBox="0 -755.9 3936.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E209-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E209-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E209-MJMAIN-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path><path stroke-width="1" id="E209-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path stroke-width="1" id="E209-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E209-MJMATHI-6E" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E209-MJMAIN-3D" x="878" y="0"></use><g transform="translate(1934,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E209-MJMAIN-37"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E209-MJMAIN-35" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E209-MJMAIN-30" x="1001" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E209-MJMAIN-30" x="1501" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-206">n = 7500</script>），如果将其两两组合作为新特征，则特征数量将为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-207-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.578ex" height="3.044ex" viewBox="0 -906.7 7568.2 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E210-MJMATHI-43" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E210-MJMATHI-69" x="5194" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E210-MJMATHI-6C" x="5539" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E210-MJMATHI-6C" x="5838" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E210-MJMATHI-69" x="6136" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E210-MJMATHI-6F" x="6482" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E210-MJMATHI-6E" x="6967" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-207">C_{2500}^{2} \approx 3\ million</script>。</p><h2><a name='header-n14' class='md-header-anchor '></a>8.2 神经网络和大脑(Neurons and the Brain)</h2><p>脑科学家通过对动物实验，发现大脑中专用于处理听觉信号的脑皮层也能处理其他诸如视觉等信号，即如果切断其与耳朵的联系，将其与眼睛相连，则这块负责听觉的脑皮层区域也能接受并处理视觉信号，从而学会“看”。脑科学家通过这类换源实验，就推论假设大脑的学习算法只有一种(“one learning algorithm” hypothesis)。那么如果能找出这种学习算法并应用于计算机中，那梦想中和人一样的人工智能就成真了。</p><p>神经网络就源于<strong>模拟人类大脑</strong>，但其需要的计算量很大。随着计算机硬件性能的提高，神经网络逐渐从衰落变为流行，如今已广泛地被应用在各行各业中。</p><p>下图是根据研究做的一些应用（有兴趣可回顾视频）：</p><p><img src='image/20180115_101441.png' alt='' /></p><p>BrainPort  系统：帮助失明人士通过摄像头以及舌尖感官“看”东西</p><p><img src='image/20180115_101442.png' alt='' /></p><p>触觉皮带：在朝北时蜂鸣器会发出声响，可使人拥有方向感（声音信号转换为方向信号）。</p><h2><a name='header-n29' class='md-header-anchor '></a>8.3 模型表示1(Model Representation I)</h2><p>既然神经网络模仿的是大脑神经元，那就先看一下大脑的神经元长什么样吧：</p><p><img src='image/20141213201613758.jpg' alt='来源: http://blog.csdn.net/zzwu/article/details/574931' /></p><p>想象一下印刷厂中流水线的工人（机器人也算哦），每个工人都有特定的任务，比如装订，塑封，贴防伪标识等等，工人们看到书本并处理完自己的任务后，就回放回传送带，紧接着传送带就传给下一个环节的工人，如此不断重复从而完成一个又一个环节，直到一本书印制完成。</p><p>那么类比一下，把上图中的<strong>细胞核(nucleus)</strong>类比成工人，<strong>轴突(axon)</strong>类比传送带，<strong>树突(dendrite)</strong>则比类比成工人的双眼。一个又一个细胞体，从树突接收需要处理的信息，对其进行处理后，再经由轴突通过电信号把处理完的信息传递出去，直到理解信息的内容。当然啦，我们大脑的实际上还要更为复杂，而且一个人的神经元数目就比地球上所有流水线的工人之和还要多呢~</p><p>人工神经网络中，树突对应<strong>输入(input)</strong>，细胞核对应<strong>激活单元(activation unit)</strong>，轴突对应<strong>输出(output)</strong>。</p><p>我们一般把神经网络划分为三部分（注意，不是只有三层！），即输入层(input layer)，隐藏层(hidden layer)和输出层(output layer)。</p><p><img src='image/20180116_001543.png' alt='' /></p><p>图中的一个圈表示神经网络中的一个激活单元，输入层对应输入单元，隐藏层对应中间单元，输出层则对应输出单元。中间激活单元应用<strong>激活函数</strong>(<a href='https://en.wikipedia.org/wiki/Activation_function'>activation_function</a>)处理数据。</p><p>下面列出一些已有概念在神经网络中的别称：</p><ul><li><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-209-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.384ex" height="1.76ex" viewBox="0 -504.6 1026.4 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E212-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E212-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 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629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E212-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E212-MJMAIN-30" x="809" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-209">x_0</script>=1</li><li><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-26-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.808ex" height="2.11ex" viewBox="0 -806.1 778.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E26-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 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width="1.128ex" height="1.41ex" viewBox="0 -504.6 485.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E219-MJMATHI-76" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E219-MJMATHI-76" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-216">v</script> 个单元的权重</p><p><span 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMAIN-31" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMAIN-2B" x="17189" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMAIN-31" x="18190" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMAIN-29" x="18691" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMAIN-3D" x="19358" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMAIN-33" x="20414" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMAIN-D7" x="21137" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E227-MJMAIN-34" x="22138" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-224">Size(\Theta^{(1)})=s_{j+1} \times (s_j + 1) =s_2 \times (s_1 + 1) = 3 \times 4</script></p><p><span 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type="math/tex" id="MathJax-Element-229">h_\Theta(x) = a_1^{(3)} = g(\Theta_{10}^{(2)}a_0^{(2)} + \Theta_{11}^{(2)}a_1^{(2)} + \Theta_{12}^{(2)}a_2^{(2)} + \Theta_{13}^{(2)}a_3^{(2)})</script></p><p>上面的计算过程被称为<strong>前向传播(Forward propagation)</strong>，即从输入层开始，一层一层地向下计算并传递结果。</p><p>&nbsp;</p><p>再回顾一下逻辑回归：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-230-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="36.531ex" height="2.577ex" viewBox="0 -806.1 15728.6 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E233-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 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x="0" y="-4285"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-233">\Theta^{(1)}=\left[\begin{matrix}\Theta^{(1)}_{10}& \Theta^{(1)}_{11}& \Theta^{(1)}_{12}& \Theta^{(1)}_{13}\\ \Theta^{(1)}_{20}& \Theta^{(1)}_{21}& \Theta^{(1)}_{22}& \Theta^{(1)}_{23}\\ \Theta^{(1)}_{30}& \Theta^{(1)}_{31}& \Theta^{(1)}_{32} & \Theta^{(1)}_{33}\end{matrix}\right]</script>，</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-234-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.752ex" height="11.216ex" viewBox="0 -2665.7 5921.1 4828.9" role="img" focusable="false" style="vertical-align: -4.874ex; margin-bottom: -0.15ex;"><defs><path stroke-width="1" id="E237-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 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xmlns:xlink="http://www.w3.org/1999/xlink" width="26.74ex" height="2.928ex" viewBox="0 -956.9 11513 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E239-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="1" id="E239-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 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xlink:href="#E241-MJMATHI-6E" x="748" y="-212"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-238">z^{(j)}_i = \Theta^{(j-1)}_{i,0}a^{(j-1)}_{0}+ \Theta^{(j-1)}_{i,1}a^{(j-1)}_{1}+\dots+ \Theta^{(j-1)}_{i,n}a^{(j-1)}_{n}</script>，</p><p> <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-239-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.995ex" height="2.461ex" viewBox="0 -956.9 7747.8 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E242-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E242-MJMATHI-6A" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E242-MJMAIN-2212" x="802" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E242-MJMAIN-31" x="1580" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E242-MJMAIN-29" x="2081" y="0"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-239">z^{(j)} = \Theta^{(j-1)}a^{(j-1)}</script>，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-240-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.722ex" height="2.928ex" viewBox="0 -956.9 5477.5 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E243-MJMATHI-61" d="M33 157Q33 258 109 349T280 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71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E243-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E243-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E243-MJMATHI-6A" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E243-MJMAIN-29" x="802" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E243-MJMAIN-3D" x="1749" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E243-MJMATHI-67" x="2806" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E243-MJMAIN-28" x="3286" y="0"></use><g transform="translate(3676,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E243-MJMATHI-7A" x="0" y="0"></use><g transform="translate(469,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E243-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E243-MJMATHI-6A" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E243-MJMAIN-29" x="802" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E243-MJMAIN-29" x="5088" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-240">a^{(j)} = g(z^{(j)})</script>，通过该式即可计算神经网络中每一层的值。</p><p>扩展到所有样本实例：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" 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xlink:href="#E244-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E244-MJMATHI-54" x="1215" y="581"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-241">{{z}^{\left( 2 \right)}}={{\Theta }^{\left( 1 \right)}} {{X}^{T}}</script>，这时 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-242-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.424ex" height="2.461ex" viewBox="0 -956.9 1474.2 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E245-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E245-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E245-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E245-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E245-MJMATHI-7A" x="0" y="0"></use><g transform="translate(469,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E245-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E245-MJMAIN-32" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E245-MJMAIN-29" x="890" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-242">z^{(2)}</script> 是一个 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-243-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.025ex" height="1.76ex" viewBox="0 -554.9 3024.9 757.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E246-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="1" id="E246-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E246-MJMAIN-D7" d="M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path><path stroke-width="1" id="E246-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E246-MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E246-MJMAIN-32" x="663" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E246-MJMAIN-D7" x="1145" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E246-MJMATHI-6D" x="2146" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-243">s_2 \times m</script> 维矩阵。</p><blockquote><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-22-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.04ex" height="1.41ex" viewBox="0 -504.6 878.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E22-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMATHI-6D" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-22">m</script>: 训练集中的样本实例数量</p><p><span 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class='md-header-anchor '></a>8.5 例子和直观理解1(Examples and Intuitions I)</h2><p>为了更好的理解神经网络，举例单层神经网络进行逻辑运算的例子。</p><p>下面的例子中，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-245-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.802ex" height="1.76ex" viewBox="0 -504.6 2498 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E248-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 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id="MathJax-Element-245">x_1,x_2</script> 为二进制数。</p><p>逻辑与(AND)运算（都为真值则结果才为真）神经网络：</p><p><img src='image/20180117_000612.png' alt='' /></p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-246-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="22.703ex" height="2.928ex" viewBox="0 -956.9 9774.9 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E249-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="1" id="E249-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 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y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E250-MJMAIN-2B" x="6784" y="0"></use><g transform="translate(7784,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E250-MJMAIN-32"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E250-MJMAIN-30" x="500" y="0"></use></g><g transform="translate(8785,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E250-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E250-MJMAIN-31" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E250-MJMAIN-2B" x="10034" y="0"></use><g transform="translate(11035,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E250-MJMAIN-32"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E250-MJMAIN-30" x="500" y="0"></use></g><g transform="translate(12036,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E251-MJMAIN-44" x="1501" y="0"></use></g><g transform="translate(7485,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E251-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E251-MJMAIN-32" x="809" y="-213"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-248">h_\theta(x)\approx x_1\ \text{AND}\ x_2</script>。这样就实现了一个能够进行与运算的神经网络。 </p><p><img src='image/2413fbec8ff9fa1f19aaf78265b8a33b_Logistic_function.png' alt='sigmoid function' /></p><p>&nbsp;</p><p>再举一例，逻辑或(OR)运算（有一个真值则结果就为真）神经网络：</p><p><img src='image/20180117_000349.png' alt='' /></p><p>&nbsp;</p><h2><a name='header-n173' class='md-header-anchor '></a>8.6 例子和直观理解2(Examples and Intuitions II)</h2><p>下面逐步构建复杂一点的神经网络</p><p><img src='image/20180117_004820.png' alt='' /></p><p>如上图，我们分别构建了三个单层神经网络，将这三个网络组合起来，可得到一个新的神经网络，其可完成逻辑运算中的异或(XNOR)操作：</p><p><img src='image/20180116_235545.png' alt='' /></p><p>这里的组合即为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-249-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="54.065ex" height="2.577ex" viewBox="0 -806.1 23278 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E252-MJMAIN-58" d="M270 0Q252 3 141 3Q46 3 31 0H23V46H40Q129 50 161 88Q165 94 244 216T324 339Q324 341 235 480T143 622Q133 631 119 634T57 637H37V683H46Q64 680 172 680Q297 680 318 683H329V637H324Q307 637 286 632T263 621Q263 618 322 525T384 431Q385 431 437 511T489 593Q490 595 490 599Q490 611 477 622T436 637H428V683H437Q455 680 566 680Q661 680 676 683H684V637H667Q585 634 551 599Q548 596 478 491Q412 388 412 387Q412 385 514 225T620 62Q628 53 642 50T695 46H726V0H717Q699 3 591 3Q466 3 445 0H434V46H440Q454 46 476 51T499 64Q499 67 463 124T390 238L353 295L350 292Q348 290 343 283T331 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a^{(3)}\end{align*}</script></p><p>可见，特征值能不断升级，并抽取出更多信息，直到计算出结果。而如此不断组合，我们就可以逐渐构造出越来越复杂、强大的神经网络，比如用于手写识别的神经网络。</p><blockquote><p>任何布尔函数都可由两层神经网络准确表达，但所需的中间单元的数量随输入呈指数级增长;</p><p>任何连续函数都可由两层神经网络以任意精度逼近;</p><p>任何函数都可由三层神经网络以任意程度逼近。</p></blockquote><h2><a name='header-n195' class='md-header-anchor '></a>8.7 多类别分类(Multiclass Classification)</h2><p>之前讨论的都是预测结果为单值情况下的神经网络，要实现多类别分类，其实只要修改一下输出层，让输出层包含多个输出单元即可。</p><p>举一个 4 分类问题的实例：</p><p><img src='image/20180117_010904.png' alt='' /></p><p>有四种分类情况，那么就让输出层包含 4 个输出单元即可，则 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-253-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.85ex" height="2.344ex" viewBox="0 -755.9 1227 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E256-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E256-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E256-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E256-MJMAIN-398" x="815" y="-219"></use></g></svg></span><script type="math/tex" id="MathJax-Element-253">h_\Theta</script> 为 4 维向量。 </p><p>神经网络中的多分类算法算是对 one-vs-all 思想的扩展，定义预测结果一共有 4 种情况：</p><p><img src='image/20180117_011331.png' alt='' /></p><p>如果预测结果 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-254-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.091ex" height="12.383ex" viewBox="0 -2917 6067.1 5331.5" role="img" focusable="false" style="vertical-align: -5.608ex;"><defs><path stroke-width="1" id="E257-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E257-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="1" id="E257-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E257-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E257-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E257-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E257-MJMAIN-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path stroke-width="1" id="E257-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E257-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E257-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path stroke-width="1" id="E257-MJSZ4-23A1" d="M319 -645V1154H666V1070H403V-645H319Z"></path><path stroke-width="1" id="E257-MJSZ4-23A3" d="M319 -644V1155H403V-560H666V-644H319Z"></path><path stroke-width="1" id="E257-MJSZ4-23A2" d="M319 0V602H403V0H319Z"></path><path stroke-width="1" id="E257-MJSZ4-23A4" d="M0 1070V1154H347V-645H263V1070H0Z"></path><path stroke-width="1" id="E257-MJSZ4-23A6" d="M263 -560V1155H347V-644H0V-560H263Z"></path><path stroke-width="1" id="E257-MJSZ4-23A5" d="M263 0V602H347V0H263Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJMAIN-398" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJMAIN-28" x="1226" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJMATHI-78" x="1616" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJMAIN-29" x="2188" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJMAIN-3D" x="2856" y="0"></use><g transform="translate(3912,0)"><g transform="translate(0,2850)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJSZ4-23A1" x="0" y="-1155"></use><g transform="translate(0,-3446.188741721854) scale(1,2.8112582781456954)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJSZ4-23A2"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJSZ4-23A3" x="0" y="-4555"></use></g><g transform="translate(834,0)"><g transform="translate(-15,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJMAIN-30" x="0" y="2050"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJMAIN-30" x="0" y="650"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJMAIN-31" x="0" y="-750"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJMAIN-30" x="0" y="-2150"></use></g></g><g transform="translate(1487,2850)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJSZ4-23A4" x="0" y="-1155"></use><g transform="translate(0,-3446.188741721854) scale(1,2.8112582781456954)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJSZ4-23A5"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E257-MJSZ4-23A6" x="0" y="-4555"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-254">h_\Theta(x) =\begin{bmatrix}0 \newline 0 \newline 1 \newline 0 \newline\end{bmatrix}</script>，</p><p>那么表示 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-255-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.043ex" height="2.577ex" viewBox="0 -806.1 3032.4 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E258-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E258-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="1" id="E258-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E258-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E258-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E258-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E258-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E258-MJMAIN-398" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E258-MJMAIN-28" x="1226" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E258-MJMATHI-78" x="1616" y="0"></use><g transform="translate(2188,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E258-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E258-MJMAIN-33" x="550" y="-213"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-255">h_\Theta(x)_3</script>，即分为第 3 类，对应于图中的摩托车(Motorcycle)。</p><p><strong>总结一下</strong>，要分为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-16-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.211ex" height="1.994ex" viewBox="0 -755.9 521.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E16-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E16-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-16">k</script> 类，就在输出层放置 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-16-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.211ex" height="1.994ex" viewBox="0 -755.9 521.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E16-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E16-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-16">k</script> 个输出单元，对于单个样本实例，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-256-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.989ex" height="2.577ex" viewBox="0 -806.1 2578.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E259-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 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314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E259-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E259-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E259-MJMAIN-398" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E259-MJMAIN-28" x="1226" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E259-MJMATHI-78" x="1616" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E259-MJMAIN-29" x="2188" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-256">h_\Theta(x)</script> 则是一个 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-16-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.211ex" height="1.994ex" viewBox="0 -755.9 521.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E16-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E16-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-16">k</script> 维结果向量，最后依据结果向量，得出属于哪个类 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-257-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.24ex" height="2.928ex" viewBox="0 -956.9 1394.8 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E260-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 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371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E260-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMATHI-79" x="0" y="0"></use><g transform="translate(499,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E260-MJMAIN-29" x="734" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-257">y^{(i)}</script>。</p><p>&nbsp;</p></div>
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diff --git a/week4.md b/week4.md
index 96753b8..cdfbd4f 100644
--- a/week4.md
+++ b/week4.md
@@ -100,7 +100,7 @@ $h_\Theta(x) = a_1^{(3)} = g(\Theta_{10}^{(2)}a_0^{(2)} + \Theta_{11}^{(2)}a_1^{
 
 ${h_\theta}\left( x \right)=g\left( {\theta_0}+{\theta_1}{x_1}+{\theta_{2}}{x_{2}}+{\theta_{3}}x_3 \right)$
 
-是不是除了符号表示，其他都完全一样？其实神经网络就好似回归模型，只不过输入变成了中间单元 $a_1^{(j)}, a_2^{(j)}, \dots, a_n^{(j)}$。从输入 $x$ 开始，下一层的每个激活单元都包含了上一层的所有信息（单元值），通过最优化算法不断迭代计算，激活单元能得出关于输入 $x$ 的更多信息，这就好像是在给假设函数加多项式。中间层的这些单元好似升级版的初始特征，从而能给出更好的预测。
+是不是除了符号表示，其他都完全一样？其实神经网络就好似回归模型，只不过输入变成了中间单元 $a_1^{(j)}, a_2^{(j)}, \dots, a_n^{(j)}$。从输入 $x$ 开始，下一层的每个激活单元都包含了上一层的所有信息（单元值），通过最优化算法不断迭代计算，激活单元能得出关于输入 $x$ 的更多信息，这就好像是在给假设函数加多项式。隐藏层的这些单元好似升级版的初始特征，从而能给出更好的预测。