From 16cd501fa1dd1405443c98032c03a6532cb56b09 Mon Sep 17 00:00:00 2001
From: scruel <scruelt@hotmail.com>
Date: Fri, 12 Jan 2018 01:18:01 +0800
Subject: [PATCH] optimize 6.7

---
 image/20180112_001720.png | Bin 68004 -> 62843 bytes
 week3.html                |   4 ++--
 week3.md                  |   2 +-
 3 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/image/20180112_001720.png b/image/20180112_001720.png
index 9af86fe25421e8c0cce0dfcbd232ad36d9a755bc..9bbb8b743da3f9da0b4595d2f0a4abc62eb80d4d 100644
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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="n1872"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n1872">6 逻辑回归(Logistic Regression)</a></span><span class="md-toc-item md-toc-h2" data-ref="n1873"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n1873">6.1 分类(Classification)</a></span><span class="md-toc-item md-toc-h2" data-ref="n1910"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n1910">6.2 假设函数表示(Hypothesis Representation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n1934"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n1934">6.3 决策边界(Decision Boundary)</a></span><span class="md-toc-item md-toc-h2" data-ref="n1979"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n1979">6.4 损失函数(Cost Function)</a></span><span class="md-toc-item md-toc-h2" data-ref="n2006"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n2006">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="n2066"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n2066">6.6 进阶优化(Advanced Optimization)</a></span><span class="md-toc-item md-toc-h2" data-ref="n2138"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n2138">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="n2156"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n2156">7 Regularization</a></span><span class="md-toc-item md-toc-h2" data-ref="n2157"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n2157">7.1 The Problem of Overfitting</a></span><span class="md-toc-item md-toc-h2" data-ref="n2158"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n2158">7.2 Cost Function</a></span><span class="md-toc-item md-toc-h2" data-ref="n2159"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n2159">7.3 Regularized Linear Regression</a></span><span class="md-toc-item md-toc-h2" data-ref="n2160"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n2160">7.4 Regularized Logistic Regression</a></span></p></div><h1><a name='header-n1872' class='md-header-anchor '></a>6 逻辑回归(Logistic Regression)</h1><h2><a name='header-n1873' 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-2-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="E2-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="E2-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 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id="E2-MJMAIN-7D" d="M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z"></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="#E2-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E2-MJMAIN-2208" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E2-MJMAIN-7B" x="1720" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E2-MJMAIN-30" x="2221" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E2-MJMAIN-2C" x="2721" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E2-MJMAIN-31" x="3166" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E2-MJMAIN-7D" x="3667" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-2"> 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-3-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" 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132Z"></path><path stroke-width="1" id="E4-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="E4-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 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y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJAMS-2A7E" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-3">h_\theta(x) \geqslant 0.5</script> ，预测为 <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="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="E62-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 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y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-61">y = 1</script>，即正向类；</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-5-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="E6-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 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-5">h_\theta(x) \lt 0.5</script> ，预测为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-59-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="E60-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="E60-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="E60-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="#E60-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E60-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E60-MJMAIN-30" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-59">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-7-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="E8-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="#E8-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-7">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 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-8-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="1.994ex" viewBox="0 -755.9 759.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E9-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 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z"></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="#E9-MJMATHI-52" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-8">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-9-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="E10-MJMATHI-68" d="M137 683Q138 683 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xlink:href="#E14-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-78" x="4928" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-13">h_\theta \left( x \right)=\theta^{T}x</script>，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-14-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="E15-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="#E15-MJMATHI-67" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-14">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-15-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.168ex" 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id="MathJax-Element-17">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-18-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" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E19-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 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E19-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E19-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E19-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E19-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-18">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-19-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="E20-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 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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="E90-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="#E90-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-89">\theta</script>，计算得出”输出 <span class="MathJax_Preview"></span><span 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x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E24-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E24-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E24-MJMAIN-3D" x="2804" y="0"></use><g transform="translate(3860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E24-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E24-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E24-MJMAIN-37" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-23">h_\theta \left( x \right)=0.7</script> 表示病人有 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-24-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.261ex" height="2.227ex" viewBox="0 -806.1 1834.5 958.9" 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMAIN-3D" x="2804" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-67" x="3860" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMAIN-28" x="4341" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-7A" x="4730" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMAIN-29" x="5199" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMAIN-3D" x="5866" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-67" x="6922" y="0"></use><g transform="translate(7569,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-78" x="1526" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJSZ1-29" x="2098" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-25">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-26-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="E27-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="E27-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="E27-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="#E27-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E27-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E27-MJMAIN-35" x="779" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-26">0.5</script> 为阈值：</p><p><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="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="E28-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="E28-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="E28-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="E28-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="E28-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="E28-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="E28-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="E28-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="E28-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="E28-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="E28-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="E28-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="E28-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="E28-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 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-15,0)"><g transform="translate(0,651)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2265" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-35" x="779" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2192" x="5251" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-79" x="6529" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-3D" x="7304" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-31" x="8361" y="0"></use></g><g transform="translate(0,-700)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-3C" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-35" x="779" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2192" x="5251" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-79" x="6529" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-3D" x="7304" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-30" x="8361" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-27">\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>回忆一下逻辑函数的图像：</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-28-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="E29-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="E29-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="E29-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="E29-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 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xlink:href="#E32-MJMAIN-2192" x="5800" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-221E" x="7078" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-21D2" x="8357" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMATHI-67" x="9635" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-28" x="10115" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMATHI-7A" x="10505" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-29" x="10973" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-3D" x="11641" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-30" x="12697" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-31">\begin{align*}z \to +\infty, e^{-\infty} \to 0 \Rightarrow g(z)=1 \newline z \to 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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="E33-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="E33-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 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y="0"></use><g transform="translate(5708,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(6632,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-29" x="7658" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-32">{h_\theta}\left( x \right)=g\left( {\theta_0}+{\theta_1}{x_1}+{\theta_{2}}{x_{2}}\right)</script> 是下图模型的假设函数：</p><p><img src='image/20180111_000814.png' alt='' /></p><p>根据上面的讨论，要进行分类，那么只要 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-33-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="21.144ex" height="2.461ex" viewBox="0 -806.1 9103.5 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E34-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="E34-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 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E62-MJMAIN-31" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-61">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-35-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="E36-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 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xlink:href="#E41-MJMAIN-2B" x="4776" y="0"></use><g transform="translate(5777,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(6701,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-2B" x="7949" y="0"></use><g transform="translate(8950,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-33" x="663" y="-213"></use></g><g transform="translate(9873,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-32" x="809" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-31" x="809" y="-435"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-2B" x="11122" y="0"></use><g transform="translate(12123,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-34" x="663" y="-213"></use></g><g transform="translate(13046,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-32" x="809" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-32" x="809" y="-435"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJSZ1-29" x="14072" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-40">{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-41-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="E42-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="E42-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="E42-MJMAIN-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path stroke-width="1" id="E42-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E42-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="E42-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMATHI-3B8" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-3D" x="747" y="0"></use><g transform="translate(1803,0)"><g transform="translate(0,3550)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJSZ4-23A1" 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="#E42-MJSZ4-23A2"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-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="#E42-MJMAIN-2212" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-31" x="778" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-30" x="389" y="1350"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-30" x="389" y="-50"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-31" x="389" y="-1450"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-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="#E42-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="#E42-MJSZ4-23A5"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJSZ4-23A6" x="0" y="-5955"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-41">\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-42-Frame" tabindex="-1" style="font-size: 100%; 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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="E43-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="E43-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="E43-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)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-31" x="809" y="-213"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-32" x="1451" y="513"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-2B" x="1702" y="0"></use><g transform="translate(2703,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-32" x="809" y="-213"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-32" x="1451" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-3D" x="4461" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-31" x="5517" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-42">{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-43-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="E44-MJMATHI-7A" d="M347 338Q337 338 294 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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="E90-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="#E90-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-89">\theta</script>)决定啦。</p><h2><a name='header-n1979' class='md-header-anchor '></a>6.4 损失函数(Cost Function)</h2><p>上节又留下了个问题，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-89-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="E90-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="#E90-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-89">\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-91-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="E92-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="E92-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 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id="MathJax-Element-47">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 <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="16.022ex" height="3.044ex" viewBox="0 -906.7 6898.4 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E70-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 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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="E70-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="E70-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></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-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMAIN-3D" x="2637" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-67" x="3694" y="0"></use><g transform="translate(4341,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-78" x="1526" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJSZ1-29" x="2098" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-69">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-91-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="E92-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="E92-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="E92-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="E92-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="#E92-MJMATHI-4A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E92-MJMAIN-28" x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E92-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E92-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-91">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，这样梯度下降算法将无法确定收敛点是全局最优。</p><p><img src='image/20180111_080514.png' alt='' /></p><p>如果是一个凸函数，可以对其进行最优化分析，这类最优化问题，称为<strong>凸优化问题</strong>。还好，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></p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-50-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="47.03ex" height="13.083ex" viewBox="0 -3067.8 20249.2 5633" role="img" focusable="false" style="vertical-align: -5.958ex;"><defs><path stroke-width="1" id="E51-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" 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x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E92-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E92-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-91">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-52-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="E53-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="E53-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="E53-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="#E53-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E53-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E53-MJMAIN-31" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-52">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-53-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="E54-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="#E54-MJMAIN-31" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-53">1</script>，损失函数的值会趋于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-60-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="E61-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="#E61-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-60">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-60-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="E61-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="#E61-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-60">0</script>，则会给出一个<strong>很高的损失</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-89-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="E90-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="#E90-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-89">\theta</script> 值，右图 <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="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="E58-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="E58-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="E58-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="#E58-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E58-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E58-MJMAIN-30" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-57">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-n2006' 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，我们把损失函数<strong>简化</strong>为一个函数： 
+<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="n293"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n293">6 逻辑回归(Logistic Regression)</a></span><span class="md-toc-item md-toc-h2" data-ref="n294"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n294">6.1 分类(Classification)</a></span><span class="md-toc-item md-toc-h2" data-ref="n331"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n331">6.2 假设函数表示(Hypothesis Representation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n355"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n355">6.3 决策边界(Decision Boundary)</a></span><span class="md-toc-item md-toc-h2" data-ref="n400"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n400">6.4 损失函数(Cost Function)</a></span><span class="md-toc-item md-toc-h2" data-ref="n427"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n427">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="n487"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n487">6.6 进阶优化(Advanced Optimization)</a></span><span class="md-toc-item md-toc-h2" data-ref="n559"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n559">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="n577"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n577">7 Regularization</a></span><span class="md-toc-item md-toc-h2" data-ref="n578"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n578">7.1 The Problem of Overfitting</a></span><span class="md-toc-item md-toc-h2" data-ref="n579"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n579">7.2 Cost Function</a></span><span class="md-toc-item md-toc-h2" data-ref="n580"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n580">7.3 Regularized Linear Regression</a></span><span class="md-toc-item md-toc-h2" data-ref="n581"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n581">7.4 Regularized Logistic Regression</a></span></p></div><h1><a name='header-n293' class='md-header-anchor '></a>6 逻辑回归(Logistic Regression)</h1><h2><a name='header-n294' 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-2-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 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747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z"></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="#E2-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E2-MJMAIN-2208" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E2-MJMAIN-7B" x="1720" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E2-MJMAIN-30" x="2221" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E2-MJMAIN-2C" x="2721" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E2-MJMAIN-31" x="3166" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E2-MJMAIN-7D" x="3667" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-2"> 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-3-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="E4-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="E4-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="E4-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="E4-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="E4-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="E4-MJAMS-2A7E" 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 616ZM674 172Q692 172 694 154Q694 142 687 137Q685 135 395 -2L107 -138H101Q83 -136 83 -118Q83 -106 96 -100Q100 -98 380 35T665 170T674 172Z"></path><path stroke-width="1" id="E4-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 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJAMS-2A7E" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-3">h_\theta(x) \geqslant 0.5</script> ，预测为 <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="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="E62-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 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y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-61">y = 1</script>，即正向类；</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-5-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="E6-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 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-5">h_\theta(x) \lt 0.5</script> ，预测为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-59-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="E60-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="E60-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="E60-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="#E60-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E60-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E60-MJMAIN-30" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-59">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-7-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="E8-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="#E8-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-7">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 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-8-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="1.994ex" viewBox="0 -755.9 759.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E9-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 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z"></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="#E9-MJMATHI-52" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-8">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-9-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="E10-MJMATHI-68" d="M137 683Q138 683 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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="#E14-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMAIN-3D" x="2804" y="0"></use><g transform="translate(3860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-78" x="4928" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-13">h_\theta \left( x \right)=\theta^{T}x</script>，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-14-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="E15-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="#E15-MJMATHI-67" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-14">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-15-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" 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E16-MJMAIN-31" x="1335" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E16-MJMAIN-29" x="1835" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-15">(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-16-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="E17-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 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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="#E20-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-19">x</script> 以及参数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-89-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="E90-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="#E90-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-89">\theta</script>，计算得出”输出 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-52-Frame" 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E23-MJMATHI-3B8" x="11620" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E23-MJMAIN-29" x="12089" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E23-MJMAIN-3D" x="12757" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E23-MJMAIN-31" x="13813" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-22">\begin{align*}& h_\theta(x) = P(y=1 | x ; \theta) = 1 - P(y=0 | x ; \theta) \newline & P(y = 0 | x;\theta) + P(y = 1 | x ; \theta) = 1\end{align*}</script></p><p>&nbsp;</p><p>以肿瘤诊断为例，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-23-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.939ex" height="2.577ex" viewBox="0 -806.1 5140.2 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path 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type="math/tex" id="MathJax-Element-24">70\%</script> 的概率得了恶性肿瘤。</p><h2><a name='header-n355' 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 class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-25-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="23.521ex" height="3.044ex" viewBox="0 -906.7 10127.1 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E26-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 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMAIN-3D" x="2804" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-67" x="3860" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMAIN-28" x="4341" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-7A" x="4730" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMAIN-29" x="5199" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMAIN-3D" x="5866" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-67" x="6922" y="0"></use><g transform="translate(7569,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJMATHI-78" x="1526" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E26-MJSZ1-29" x="2098" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-25">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-26-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="E27-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="E27-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="E27-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="#E27-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E27-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E27-MJMAIN-35" x="779" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-26">0.5</script> 为阈值：</p><p><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="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="E28-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="E28-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="E28-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="E28-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="E28-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 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id="E28-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="E28-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="E28-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="E28-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="E28-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="E28-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="E28-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 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-15,0)"><g transform="translate(0,651)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2265" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-35" x="779" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2192" x="5251" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-79" x="6529" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-3D" x="7304" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-31" x="8361" y="0"></use></g><g transform="translate(0,-700)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-3C" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-35" x="779" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2192" x="5251" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-79" x="6529" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-3D" x="7304" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-30" x="8361" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-27">\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>回忆一下逻辑函数的图像：</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-28-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="E29-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="E29-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="E29-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="E29-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 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id="E29-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="#E29-MJMATHI-67" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E29-MJMAIN-28" x="480" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E29-MJMATHI-7A" x="870" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E29-MJMAIN-29" x="1338" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E29-MJMAIN-2265" x="2005" y="0"></use><g transform="translate(3062,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E29-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E29-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E29-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-28">g(z) \geq 0.5</script> 时，有 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-37-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="E38-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 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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="E31-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="E31-MJMAIN-2265" d="M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 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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="E32-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="E32-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 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y="0"></use></g><g transform="translate(50,-700)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMATHI-7A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-2192" x="746" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-2212" x="2024" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-221E" x="2803" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-2C" x="3803" y="0"></use><g transform="translate(4248,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMATHI-65" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-221E" x="659" y="583"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-2192" x="5800" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-221E" x="7078" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-21D2" x="8357" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMATHI-67" x="9635" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-28" x="10115" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMATHI-7A" x="10505" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-29" x="10973" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-3D" x="11641" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E32-MJMAIN-30" x="12697" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-31">\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 class="MathJax_Preview"></span><span class="MathJax_SVG" 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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="E33-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="E33-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 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y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-30" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-2B" x="1535" y="0"></use><g transform="translate(2535,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-31" x="663" y="-213"></use></g><g transform="translate(3459,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-31" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-2B" x="4707" y="0"></use><g transform="translate(5708,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(6632,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E33-MJMAIN-29" x="7658" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-32">{h_\theta}\left( x \right)=g\left( {\theta_0}+{\theta_1}{x_1}+{\theta_{2}}{x_{2}}\right)</script> 是下图模型的假设函数：</p><p><img src='image/20180111_000814.png' alt='' /></p><p>根据上面的讨论，要进行分类，那么只要 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-33-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="21.144ex" height="2.461ex" viewBox="0 -806.1 9103.5 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y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-61">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-35-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="E36-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="E36-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 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y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-31" x="809" y="-435"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-2B" x="11122" y="0"></use><g transform="translate(12123,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-34" x="663" y="-213"></use></g><g transform="translate(13046,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-32" x="809" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJMAIN-32" x="809" y="-435"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E41-MJSZ1-29" x="14072" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-40">{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-41-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="E42-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 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xlink:href="#E42-MJMAIN-3D" x="747" y="0"></use><g transform="translate(1803,0)"><g transform="translate(0,3550)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJSZ4-23A1" 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="#E42-MJSZ4-23A2"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-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="#E42-MJMAIN-2212" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-31" x="778" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-30" x="389" y="1350"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-30" x="389" y="-50"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-31" x="389" y="-1450"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-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="#E42-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="#E42-MJSZ4-23A5"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJSZ4-23A6" x="0" y="-5955"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-41">\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-42-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 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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)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-31" x="809" y="-213"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-32" x="1451" y="513"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-2B" x="1702" y="0"></use><g transform="translate(2703,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-32" x="809" y="-213"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMAIN-32" x="1451" y="513"></use></g><use 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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="#E44-MJMATHI-7A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-43">z</script> (中的 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-89-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="E90-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="#E90-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-89">\theta</script>)决定啦。</p><h2><a name='header-n400' class='md-header-anchor '></a>6.4 损失函数(Cost Function)</h2><p>上节又留下了个问题，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-89-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="E90-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="#E90-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-89">\theta</script> 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\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 <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="16.022ex" height="3.044ex" viewBox="0 -906.7 6898.4 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E70-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 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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="E70-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="E70-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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xlink:href="#E70-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMAIN-3D" x="2637" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-67" x="3694" y="0"></use><g transform="translate(4341,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-78" x="1526" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJSZ1-29" x="2098" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-69">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-91-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="E92-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 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/></p><p>如果是一个凸函数，可以对其进行最优化分析，这类最优化问题，称为<strong>凸优化问题</strong>。还好，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></p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-50-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="47.03ex" height="13.083ex" viewBox="0 -3067.8 20249.2 5633" role="img" focusable="false" style="vertical-align: -5.958ex;"><defs><path stroke-width="1" id="E51-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="E51-MJMAIN-28" d="M94 250Q94 319 104 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E92-MJMATHI-3B8" x="1023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E92-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-91">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-52-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="E53-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="E53-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="E53-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="#E53-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E53-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E53-MJMAIN-31" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-52">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-53-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="E54-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="#E54-MJMAIN-31" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-53">1</script>，损失函数的值会趋于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-60-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="E61-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="#E61-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-60">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-60-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="E61-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="#E61-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-60">0</script>，则会给出一个<strong>很高的损失</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-89-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="E90-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="#E90-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-89">\theta</script> 值，右图 <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="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="E58-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 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y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMAIN-2212" x="15450" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMAIN-28" x="16451" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMAIN-31" x="16840" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMAIN-2212" x="17563" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMATHI-79" x="18564" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMAIN-29" x="19061" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMAIN-D7" x="19673" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMATHI-6C" x="20674" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMATHI-6F" x="20972" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMATHI-67" x="21458" y="0"></use><g transform="translate(22105,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMAIN-31" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMAIN-2212" x="1112" y="0"></use><g transform="translate(2112,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMATHI-3B8" x="815" y="-219"></use></g><g transform="translate(3288,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E59-MJMAIN-29" x="4639" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-58">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-59-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="E60-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 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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="#E60-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E60-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E60-MJMAIN-30" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-59">y = 0</script>，左边式子整体为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-60-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="E61-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 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y="-430"></use></g></g></g><g transform="translate(19653,0)"><g transform="translate(0,66)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-66"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-6F" x="306" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-72" x="807" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-6A" x="1449" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-3A" x="2006" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-3D" x="2284" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-30" x="3313" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-2C" x="3813" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-31" x="4092" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" 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26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E70-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="E70-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 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMAIN-3D" x="2637" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-67" x="3694" y="0"></use><g transform="translate(4341,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-78" x="1526" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJSZ1-29" x="2098" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-69">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" 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325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z"></path><path stroke-width="1" id="E73-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="E73-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" 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xlink:href="#E73-MJSZ1-28"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-31" x="458" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-2212" x="1181" y="0"></use><g transform="translate(2181,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-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="#E73-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJSZ1-29" x="3576" y="-1"></use></g><g transform="translate(15875,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-6C"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-6F" x="278" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-67" x="779" y="0"></use></g><g transform="translate(17155,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJSZ1-28"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-31" x="458" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-2212" x="1181" y="0"></use><g transform="translate(2181,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-3B8" x="815" y="-219"></use></g><g transform="translate(3357,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-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="#E73-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJSZ1-29" x="1926" y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJSZ1-29" x="5741" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-72">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>将 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-73-Frame" tabindex="-1" style="font-size: 100%; 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJMAIN-6C"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJMAIN-6F" x="278" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJMAIN-67" x="779" y="0"></use></g><g transform="translate(17208,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJSZ2-28"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJMAIN-31" x="597" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJMAIN-2B" x="1320" y="0"></use><g transform="translate(2320,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJMATHI-54" x="663" y="612"></use><g transform="translate(754,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJMATHI-78" x="0" y="0"></use><g transform="translate(404,256)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJMAIN-29" x="735" y="0"></use></g></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E76-MJSZ2-29" x="4680" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-75">=-{{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)</script></p><p>根据求偏导的性质，没有 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-76-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2ex" height="2.694ex" viewBox="0 -806.1 861.2 1160" role="img" focusable="false" style="vertical-align: -0.822ex;"><defs><path stroke-width="1" id="E77-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="E77-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="#E77-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E77-MJMATHI-6A" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-76">\theta_j</script> 的项都消去，则得：</p><p><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="17.708ex" height="4.212ex" viewBox="0 -1107.7 7624.2 1813.3" role="img" focusable="false" style="vertical-align: -1.639ex;"><defs><path stroke-width="1" id="E78-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="E78-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="E78-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><path stroke-width="1" id="E78-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="E78-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="E78-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="E78-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="E78-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="E78-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="E78-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 stroke-width="1" id="E78-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="1130" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJMAIN-2202" x="515" y="594"></use><g transform="translate(60,-411)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJMAIN-2202" x="0" y="0"></use><g transform="translate(401,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJMATHI-6A" x="663" y="-213"></use></g></g></g><g transform="translate(1370,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJMATHI-54" x="663" y="513"></use></g><g transform="translate(1526,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-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="#E78-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJSZ1-29" x="2993" y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJMAIN-3D" x="5100" y="0"></use><g transform="translate(6156,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-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="#E78-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJMAIN-29" x="734" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E78-MJMATHI-6A" x="809" y="-430"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-77">\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-78-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="65.095ex" height="4.445ex" viewBox="0 -1208.2 28026.8 1913.9" role="img" focusable="false" style="vertical-align: -1.639ex;"><defs><path stroke-width="1" id="E79-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="E79-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="E79-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><path stroke-width="1" id="E79-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="E79-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="E79-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="E79-MJMAIN-3D" d="M56 347Q56 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E92-MJMAIN-29" x="1492" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-91">J(\theta)</script> 值。</p><p>我们编写代码给出损失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损失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-92-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" 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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-93-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="E94-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="E94-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E94-MJMAIN-32" x="550" y="513"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-93">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-94-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="E95-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="E95-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="E95-MJMAIN-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path stroke-width="1" id="E95-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="E95-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="E95-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path stroke-width="1" id="E95-MJSZ3-5B" d="M247 -949V1450H516V1388H309V-887H516V-949H247Z"></path><path stroke-width="1" id="E95-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="#E95-MJMATHI-3B8" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E95-MJMAIN-3D" x="747" y="0"></use><g transform="translate(1803,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E95-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="#E95-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E95-MJMAIN-31" x="663" y="-213"></use></g><g transform="translate(0,-750)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E95-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E95-MJMAIN-32" x="663" y="-213"></use></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E95-MJSZ3-5D" x="1770" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-94">\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损失函数及其偏导数：</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"><span><span>​</span>x</span></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: 90px;"></div><div class="CodeMirror-gutters" style="height: 120px;"><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>: 标记损失函数是否收敛</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-n2138' 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）。(⊙﹏⊙)有点累，让我歇歇，丢张图去休息啦。。。</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-95-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" 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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="#E98-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E98-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E98-MJMATHI-69" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-97">y=i</script>（属于第 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-98-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="E99-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="#E99-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-98">i</script> 个分类）的可能性</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" 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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="E101-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="E101-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="#E101-MJMATHI-6B" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMAIN-3D" x="799" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMAIN-33" x="1855" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-100">k=3</script>。</p></blockquote><p>注意多类别分类问题中 <span 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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="#E105-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-104">k</script>，现在 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-103-Frame" tabindex="-1" style="font-size: 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E88-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E88-MJMAIN-29" x="734" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E88-MJMATHI-6A" x="809" y="-430"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-87">\frac{\partial }{\partial {\theta_{j}}}J(\theta) = -\frac{1}{m}\sum\limits_{i=1}^{m}{\frac{\partial }{\partial {\theta_{j}}}f(\theta)}=\frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_j^{(i)} </script></p><h2><a name='header-n487' class='md-header-anchor '></a>6.6 进阶优化(Advanced Optimization)</h2><p>运行梯度下降算法，其能最小化损失函数 <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="4.371ex" 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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="E92-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="#E92-MJMATHI-4A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E92-MJMAIN-28" x="633" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E92-MJMATHI-3B8" x="1023" y="0"></use><use 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E94-MJMAIN-32" x="550" y="513"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-93">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-94-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="E95-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="E95-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="E95-MJMAIN-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path stroke-width="1" id="E95-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="E95-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="E95-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path stroke-width="1" id="E95-MJSZ3-5B" d="M247 -949V1450H516V1388H309V-887H516V-949H247Z"></path><path stroke-width="1" id="E95-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="#E95-MJMATHI-3B8" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E95-MJMAIN-3D" x="747" y="0"></use><g transform="translate(1803,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E95-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="#E95-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E95-MJMAIN-31" x="663" y="-213"></use></g><g transform="translate(0,-750)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E95-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E95-MJMAIN-32" x="663" y="-213"></use></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E95-MJSZ3-5D" x="1770" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-94">\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损失函数及其偏导数：</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"><span><span>​</span>x</span></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: 90px;"></div><div class="CodeMirror-gutters" style="height: 120px;"><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>: 标记损失函数是否收敛</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-n559' 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）。(⊙﹏⊙)有点累，让我歇歇，丢张图去休息啦。。。</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-95-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" 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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="#E98-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E98-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E98-MJMATHI-69" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-97">y=i</script>（属于第 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-98-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="E99-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="#E99-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-98">i</script> 个分类）的可能性</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-104-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="E105-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 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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="E101-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="E101-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="#E101-MJMATHI-6B" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMAIN-3D" x="799" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMAIN-33" x="1855" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-100">k=3</script>。</p></blockquote><p>注意多类别分类问题中 <span 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y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-29" x="1970" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-103">h_\theta(x)</script> 的结果不再只是一个实数而是一个向量，如果类别总数为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-104-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="E105-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="#E105-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-104">k</script>，现在 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-103-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" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E104-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="E104-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="E104-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="E104-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="E104-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="#E104-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-28" x="1008" y="0"></use><use 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xlink:href="#E107-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E107-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-105">y = \mathop{\max}\limits_i\,h_\theta^{\left( i \right)}\left( x \right)</script>。</p><h1><a name='header-n577' class='md-header-anchor '></a>7 Regularization</h1><h2><a name='header-n578' class='md-header-anchor '></a>7.1 The Problem of Overfitting</h2><h2><a name='header-n579' class='md-header-anchor '></a>7.2 Cost Function</h2><h2><a name='header-n580' class='md-header-anchor '></a>7.3 Regularized Linear Regression</h2><h2><a name='header-n581' class='md-header-anchor '></a>7.4 Regularized Logistic Regression</h2><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></div>
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diff --git a/week3.md b/week3.md
index 9006e12..8c85c03 100644
--- a/week3.md
+++ b/week3.md
@@ -291,7 +291,7 @@ exitFlag = 1
 
 注意多类别分类问题中 $h_\theta(x)$ 的结果不再只是一个实数而是一个向量，如果类别总数为 $k$，现在 $h_\theta(x)$ 就是一个 $k$ 维向量。
 
-对于样本实例来说，只要看分为哪个类别时预测输出的值最大，就说它输出属于哪个类别，即 $y = \mathop{\max}\limits_i\,h_\theta^{\left( i \right)}\left( x \right)$。
+对于某个样本实例，需计算所有的 $k$ 种分类情况得到 $h_\theta(x)$，然后看分为哪个类别时预测输出的值最大，就说它输出属于哪个类别，即 $y = \mathop{\max}\limits_i\,h_\theta^{\left( i \right)}\left( x \right)$。
 
 # 7 Regularization
 ## 7.1 The Problem of Overfitting