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许鸣辉/ML-AndrewNg-Notes

forked from scruel/Notes-ML-AndrewNg 
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week3.html 776.51 KB
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scruel 提交于 2019-05-25 13:51 . images path in htmls
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<div  id='write'  class = 'is-node'><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" href="#header-n2">6 逻辑回归(Logistic Regression)</a></span><span class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" href="#header-n3">6.1 分类(Classification)</a></span><span class="md-toc-item md-toc-h2" data-ref="n24"><a class="md-toc-inner" href="#header-n24">6.2 假设函数表示(Hypothesis Representation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n38"><a class="md-toc-inner" href="#header-n38">6.3 决策边界(Decision Boundary)</a></span><span class="md-toc-item md-toc-h2" data-ref="n61"><a class="md-toc-inner" href="#header-n61">6.4 代价函数(Cost Function)</a></span><span class="md-toc-item md-toc-h2" data-ref="n76"><a class="md-toc-inner" href="#header-n76">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="n103"><a class="md-toc-inner" href="#header-n103">6.6 进阶优化(Advanced Optimization)</a></span><span class="md-toc-item md-toc-h2" data-ref="n148"><a class="md-toc-inner" href="#header-n148">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="n158"><a class="md-toc-inner" href="#header-n158">7 正则化(Regularization)</a></span><span class="md-toc-item md-toc-h2" data-ref="n159"><a class="md-toc-inner" href="#header-n159">7.1 过拟合问题(The Problem of Overfitting)</a></span><span class="md-toc-item md-toc-h2" data-ref="n205"><a class="md-toc-inner" href="#header-n205">7.2 代价函数(Cost Function)</a></span><span class="md-toc-item md-toc-h2" data-ref="n241"><a class="md-toc-inner" href="#header-n241">7.3 线性回归正则化(Regularized Linear Regression)</a></span><span class="md-toc-item md-toc-h2" data-ref="n258"><a class="md-toc-inner" href="#header-n258">7.4 逻辑回归正则化(Regularized Logistic Regression)</a></span></p></div><h1><a name='header-n2' class='md-header-anchor '></a>6 逻辑回归(Logistic Regression)</h1><h2><a name='header-n3' class='md-header-anchor '></a>6.1 分类(Classification)</h2><p>在分类问题中，预测的结果是离散值（结果是否属于某一类），逻辑回归算法(Logistic Regression)被用于解决这类分类问题。</p><ul><li>垃圾邮件判断</li><li>金融欺诈判断</li><li>肿瘤诊断</li></ul><p>讨论肿瘤诊断问题：</p><p><img src='images/20180109_144040.png' alt='' referrerPolicy='no-referrer' /></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_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.672ex" height="2.577ex" viewBox="0 -806.1 4164.2 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E149-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 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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 xlink:href="#E149-MJMATHI-79" x="0" y="0"></use><use xlink:href="#E149-MJMAIN-2208" x="774" y="0"></use><use xlink:href="#E149-MJMAIN-7B" x="1719" y="0"></use><use xlink:href="#E149-MJMAIN-30" x="2219" y="0"></use><use xlink:href="#E149-MJMAIN-2C" x="2719" y="0"></use><use xlink:href="#E149-MJMAIN-31" x="3164" y="0"></use><use xlink:href="#E149-MJMAIN-7D" x="3664" y="0"></use></g></svg></span><script type="math/tex"> 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_SVG" tabindex="-1" style="font-size: 100%; display: 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x="1396" y="0"></use><use xlink:href="#E152-MJMAIN-29" x="1968" y="0"></use><use xlink:href="#E152-MJMAIN-3C" x="2635" y="0"></use><g transform="translate(3691,0)"><use xlink:href="#E152-MJMAIN-30"></use><use xlink:href="#E152-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E152-MJMAIN-35" x="778" y="0"></use></g></g></svg></span><script type="math/tex">h_\theta(x) \lt 0.5</script> ，预测为 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.413ex" height="2.461ex" viewBox="0 -755.9 2330.6 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E199-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 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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 xlink:href="#E199-MJMATHI-79" x="0" y="0"></use><use xlink:href="#E199-MJMAIN-3D" x="774" y="0"></use><use xlink:href="#E199-MJMAIN-30" x="1830" y="0"></use></g></svg></span><script type="math/tex">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_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.154ex" height="1.877ex" viewBox="0 -504.6 497 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E27-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E27-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex">y</script>。</p><p>&nbsp;</p><p>接下来加入偏差项，线性回归算法给出了靛青色的拟合直线，如果阈值仍然为 0.5，可以看到算法在某些情况下会给出完全错误的结果，对于癌症、肿瘤诊断这类要求预测极其精确的问题，这种情况是无法容忍的。</p><p>不仅如此，线性回归算法的值域为全体实数集（<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.078ex" height="2.577ex" viewBox="0 -806.1 4339.2 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E154-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="0" id="E154-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E154-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E154-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="0" id="E154-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E154-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path stroke-width="0" id="E154-MJMATHI-52" d="M230 637Q203 637 198 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10000, h_\theta(x) = -10000</script> 等很大/很小(负数)的数值时，结果 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.672ex" height="2.577ex" viewBox="0 -806.1 4164.2 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E156-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 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x="2096" y="-1"></use></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-147">h_\theta \left( x \right)=g(z)=g\left(\theta^{T}x \right)</script></div></div><p>对比线性回归函数 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.766ex" height="2.811ex" viewBox="0 -906.7 5496.7 1210.2" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E159-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 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y="0"></use><use xlink:href="#E159-MJMATHI-78" x="389" y="0"></use><use xlink:href="#E159-MJMAIN-29" x="961" y="0"></use></g><use xlink:href="#E159-MJMAIN-3D" x="2802" y="0"></use><g transform="translate(3857,0)"><use xlink:href="#E159-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E159-MJMATHI-54" x="663" y="513"></use></g><use xlink:href="#E159-MJMATHI-78" x="4924" y="0"></use></g></svg></span><script type="math/tex">h_\theta \left( x \right)=\theta^{T}x</script>，<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.115ex" height="1.877ex" viewBox="0 -504.6 480 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E160-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 xlink:href="#E160-MJMATHI-67" x="0" y="0"></use></g></svg></span><script type="math/tex">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_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.162ex" height="2.577ex" viewBox="0 -806.1 2222.7 1109.7" role="img" focusable="false" 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y="0"></use></g></svg></span><script type="math/tex">(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_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.805ex" height="3.511ex" viewBox="0 -956.9 5513.1 1511.8" role="img" focusable="false" style="vertical-align: -1.289ex;"><defs><path stroke-width="0" id="E162-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 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y="0"></use></g><use xlink:href="#E167-MJMAIN-3D" x="2802" y="0"></use><g transform="translate(3857,0)"><use xlink:href="#E167-MJMAIN-30"></use><use xlink:href="#E167-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E167-MJMAIN-37" x="778" y="0"></use></g></g></svg></span><script type="math/tex">h_\theta \left( x \right)=0.7</script> 表示病人有 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.257ex" height="2.227ex" viewBox="0 -806.1 1833 958.9" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="0" id="E168-MJMAIN-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 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737 292T776 146T737 0T646 -56Q590 -56 545 0T500 146ZM651 -18Q679 -18 707 24T736 146Q736 215 711 262T644 309Q637 309 630 306T611 295T591 265T578 212Q577 200 577 146V124Q577 -18 647 -18H651Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E168-MJMAIN-37"></use><use xlink:href="#E168-MJMAIN-30" x="500" y="0"></use><use xlink:href="#E168-MJMAIN-25" x="1000" y="0"></use></g></svg></span><script type="math/tex">70\%</script> 的概率得了恶性肿瘤。</p><h2><a name='header-n38' 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_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="23.502ex" height="3.044ex" viewBox="0 -906.7 10118.9 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="0" id="E169-MJMATHI-68" d="M137 683Q138 683 209 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id="E170-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="0" id="E170-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 xlink:href="#E170-MJMAIN-30"></use><use xlink:href="#E170-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E170-MJMAIN-35" x="778" y="0"></use></g></svg></span><script type="math/tex">0.5</script> 为阈值：</p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="21.308ex" height="5.846ex" viewBox="0 -1509.8 9174.4 2517" role="img" focusable="false" style="vertical-align: -2.339ex;"><defs><path stroke-width="0" id="E171-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="0" id="E171-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="0" id="E171-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="0" id="E171-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="0" id="E171-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="0" id="E171-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="0" id="E171-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="0" id="E171-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="0" id="E171-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="0" id="E171-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="0" id="E171-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="0" id="E171-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="0" id="E171-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="0" id="E171-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(0,650)"><use xlink:href="#E171-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E171-MJMATHI-3B8" x="814" y="-218"></use><use xlink:href="#E171-MJMAIN-28" x="1007" y="0"></use><use xlink:href="#E171-MJMATHI-78" x="1396" y="0"></use><use xlink:href="#E171-MJMAIN-29" x="1968" y="0"></use><use xlink:href="#E171-MJMAIN-2265" x="2635" y="0"></use><g transform="translate(3691,0)"><use xlink:href="#E171-MJMAIN-30"></use><use xlink:href="#E171-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E171-MJMAIN-35" x="778" y="0"></use></g><use xlink:href="#E171-MJMAIN-2192" x="5246" y="0"></use><use xlink:href="#E171-MJMATHI-79" x="6524" y="0"></use><use xlink:href="#E171-MJMAIN-3D" x="7299" y="0"></use><use xlink:href="#E171-MJMAIN-31" x="8355" y="0"></use></g><g transform="translate(0,-700)"><use xlink:href="#E171-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E171-MJMATHI-3B8" x="814" y="-218"></use><use xlink:href="#E171-MJMAIN-28" x="1007" y="0"></use><use xlink:href="#E171-MJMATHI-78" x="1396" y="0"></use><use xlink:href="#E171-MJMAIN-29" x="1968" y="0"></use><use xlink:href="#E171-MJMAIN-3C" x="2635" y="0"></use><g transform="translate(3691,0)"><use xlink:href="#E171-MJMAIN-30"></use><use xlink:href="#E171-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E171-MJMAIN-35" x="778" y="0"></use></g><use xlink:href="#E171-MJMAIN-2192" x="5246" y="0"></use><use xlink:href="#E171-MJMATHI-79" x="6524" y="0"></use><use xlink:href="#E171-MJMAIN-3D" x="7299" y="0"></use><use xlink:href="#E171-MJMAIN-30" x="8355" y="0"></use></g></g></g></g></svg></span><script type="math/tex">\begin{align*}& h_\theta(x) \geq 0.5 \rightarrow y = 1 \newline& h_\theta(x) < 0.5 \rightarrow y = 0 \newline\end{align*}</script></p><p>回忆一下 sigmoid 函数的图像：</p><p><img src='images/2413fbec8ff9fa1f19aaf78265b8a33b_Logistic_function.png' alt='sigmoid function' referrerPolicy='no-referrer' /></p><p>观察可得当 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.074ex" height="2.577ex" viewBox="0 -806.1 4337.6 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E172-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="0" id="E172-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="0" id="E172-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="0" id="E172-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="0" id="E172-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="0" id="E172-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="0" id="E172-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="0" id="E172-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 xlink:href="#E172-MJMATHI-67" x="0" y="0"></use><use xlink:href="#E172-MJMAIN-28" x="480" y="0"></use><use xlink:href="#E172-MJMATHI-7A" x="869" y="0"></use><use xlink:href="#E172-MJMAIN-29" x="1337" y="0"></use><use xlink:href="#E172-MJMAIN-2265" x="2003" y="0"></use><g transform="translate(3059,0)"><use xlink:href="#E172-MJMAIN-30"></use><use xlink:href="#E172-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E172-MJMAIN-35" x="778" y="0"></use></g></g></svg></span><script type="math/tex">g(z) \geq 0.5</script> 时，有 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.346ex" height="2.232ex" viewBox="0 -757.4 2301.6 960.8" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E181-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="0" id="E181-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="0" id="E181-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 xlink:href="#E181-MJMATHI-7A" x="0" y="0"></use><use xlink:href="#E181-MJMAIN-2265" x="745" y="0"></use><use xlink:href="#E181-MJMAIN-30" x="1801" y="0"></use></g></svg></span><script type="math/tex">z \geq 0</script>，即 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.065ex" height="2.577ex" viewBox="0 -906.7 3472.4 1109.7" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E174-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="0" id="E174-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="0" id="E174-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="0" id="E174-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="0" id="E174-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 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y="0"></use><use xlink:href="#E177-MJMAIN-30" x="8597" y="0"></use></g></svg></span><script type="math/tex"> {\theta_0}+{\theta_1}{x_1}+{\theta_{2}}{x_{2}}\geq0</script> 时，就预测 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.413ex" height="2.461ex" viewBox="0 -755.9 2330.6 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E200-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 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xlink:href="#E200-MJMAIN-31" x="1830" y="0"></use></g></svg></span><script type="math/tex">y = 1</script>，即预测为正向类。</p><p>如果取 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.994ex" height="9.114ex" viewBox="0 -2213.4 4733.6 3924.2" role="img" focusable="false" style="vertical-align: -3.974ex;"><defs><path stroke-width="0" id="E179-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="0" id="E179-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 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x="0" y="-3156"></use></g></g></g></svg></span><script type="math/tex">\theta = \begin{bmatrix} -3\\1\\1\end{bmatrix}</script>，则有 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.595ex" height="2.232ex" viewBox="0 -757.4 7575.6 960.8" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E180-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 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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="0" id="E181-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 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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 xlink:href="#E195-MJMATHI-79" x="0" y="0"></use><use xlink:href="#E195-MJMAIN-3D" x="774" y="0"></use><use xlink:href="#E195-MJMAIN-31" x="1830" y="0"></use></g></svg></span><script type="math/tex">y=1</script> 的分类预测结果。</p><p>&nbsp;</p><p>上面讨论了逻辑回归模型中线性拟合的例子，下面则是一个多项式拟合的例子，和线性回归中的情况也是类似的。</p><p>为了拟合下图数据，建模多项式假设函数：</p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="44.188ex" height="3.044ex" viewBox="0 -906.7 19025.3 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="0" id="E184-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 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y="0"></use><use transform="scale(0.707)" xlink:href="#E186-MJMAIN-31" x="808" y="-213"></use><use transform="scale(0.707)" xlink:href="#E186-MJMAIN-32" x="1450" y="513"></use><use xlink:href="#E186-MJMAIN-2B" x="1701" y="0"></use><g transform="translate(2701,0)"><use xlink:href="#E186-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E186-MJMAIN-32" x="808" y="-213"></use><use transform="scale(0.707)" xlink:href="#E186-MJMAIN-32" x="1450" y="513"></use></g><use xlink:href="#E186-MJMAIN-3D" x="4458" y="0"></use><use xlink:href="#E186-MJMAIN-31" x="5514" y="0"></use></g></svg></span><script type="math/tex">{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='images/20180111_000653.png' alt='' referrerPolicy='no-referrer' /></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_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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y="0"></use></g></svg></span><script type="math/tex">z</script> (中的 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script>)决定啦。</p><h2><a name='header-n61' class='md-header-anchor '></a>6.4 代价函数(Cost Function)</h2><p>那我们怎么知道决策边界是啥样？<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\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_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.366ex" height="2.577ex" viewBox="0 -806.1 1880 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E230-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="0" id="E230-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 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transform="translate(499,362)"><use transform="scale(0.707)" xlink:href="#E189-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E189-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E189-MJMAIN-29" x="733" y="0"></use></g></g><use xlink:href="#E189-MJSZ1-29" x="6630" y="-1"></use><use transform="scale(0.707)" xlink:href="#E189-MJMAIN-32" x="10024" y="877"></use></g></g></svg></span><script type="math/tex">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_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.009ex" height="3.044ex" viewBox="0 -906.7 6892.7 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="0" id="E206-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 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405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path stroke-width="0" id="E206-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 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="0" id="E206-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="0" id="E206-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 xlink:href="#E206-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E206-MJMATHI-3B8" x="814" y="-218"></use><use xlink:href="#E206-MJMAIN-28" x="1007" y="0"></use><use xlink:href="#E206-MJMATHI-78" x="1396" y="0"></use><use xlink:href="#E206-MJMAIN-29" x="1968" y="0"></use><use xlink:href="#E206-MJMAIN-3D" x="2635" y="0"></use><use xlink:href="#E206-MJMATHI-67" x="3691" y="0"></use><g transform="translate(4337,0)"><use xlink:href="#E206-MJSZ1-28"></use><g transform="translate(458,0)"><use xlink:href="#E206-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E206-MJMATHI-54" x="663" y="513"></use></g><use xlink:href="#E206-MJMATHI-78" x="1524" y="0"></use><use xlink:href="#E206-MJSZ1-29" x="2096" y="-1"></use></g></g></svg></span><script type="math/tex">h_\theta(x) = g\left(\theta^{T}x \right)</script>，可绘制关于 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.366ex" height="2.577ex" viewBox="0 -806.1 1880 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E230-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="0" id="E230-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="0" id="E230-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="0" id="E230-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 xlink:href="#E230-MJMATHI-4A" x="0" y="0"></use><use xlink:href="#E230-MJMAIN-28" x="633" y="0"></use><use xlink:href="#E230-MJMATHI-3B8" x="1022" y="0"></use><use xlink:href="#E230-MJMAIN-29" x="1491" y="0"></use></g></svg></span><script type="math/tex">J(\theta)</script> 的图像，如下图</p><p><img src='images/20180111_080314.png' alt='' referrerPolicy='no-referrer' /></p><p>回忆线性回归中的平方损失函数，其是一个二次凸函数（碗状），二次凸函数的重要性质是只有一个局部最小点即全局最小点。上图中有许多局部最小点，这样将使得梯度下降算法无法确定收敛点是全局最优。</p><p><img src='images/20180111_080514.png' alt='' referrerPolicy='no-referrer' /></p><p>如果此处的损失函数也是一个凸函数，是否也有同样的性质，从而最优化？这类讨论凸函数最优值的问题，被称为<strong>凸优化问题(Convex optimization)</strong>。</p><p>当然，损失函数不止平方损失函数一种。</p><p>对于逻辑回归，更换平方损失函数为<strong>对数损失函数</strong>，可由统计学中的最大似然估计方法推出代价函数 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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y="0"></use></g></svg></span><script type="math/tex">J(\theta)</script>：</p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="46.994ex" height="13.083ex" viewBox="0 -3067.8 20233.5 5633" role="img" focusable="false" style="vertical-align: -5.958ex;"><defs><path stroke-width="0" id="E193-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="0" id="E193-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 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style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.161ex" height="1.877ex" viewBox="0 -755.9 500 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E196-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 xlink:href="#E196-MJMAIN-31" x="0" y="0"></use></g></svg></span><script type="math/tex">1</script>，代价函数的值会趋于 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.161ex" height="1.994ex" viewBox="0 -755.9 500 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E45-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E45-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex">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_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.161ex" height="1.994ex" viewBox="0 -755.9 500 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E45-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E45-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex">0</script>，则会给出一个<strong>很高的代价</strong>，拟合程度<strong>差</strong>，算法会根据其迅速纠正 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script> 值，右图 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.413ex" height="2.461ex" viewBox="0 -755.9 2330.6 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E197-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 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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 xlink:href="#E197-MJMATHI-79" x="0" y="0"></use><use xlink:href="#E197-MJMAIN-3D" x="774" y="0"></use><use xlink:href="#E197-MJMAIN-30" x="1830" y="0"></use></g></svg></span><script type="math/tex">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-n76' class='md-header-anchor '></a>6.5 简化的成本函数和梯度下降(Simplified Cost Function and Gradient Descent)</h2><p>由于懒得分类讨论，对于二元分类问题，我们可把代价函数<strong>简化</strong>为一个函数： 

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3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path stroke-width="0" id="E205-MJMAIN-69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z"></path><path stroke-width="0" id="E205-MJMAIN-6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 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x="4476" y="0"></use><use xlink:href="#E205-MJMAIN-6C" x="4754" y="0"></use><use xlink:href="#E205-MJMAIN-63" x="5282" y="0"></use><use xlink:href="#E205-MJMAIN-6F" x="5726" y="0"></use><use xlink:href="#E205-MJMAIN-6E" x="6226" y="0"></use><use xlink:href="#E205-MJMAIN-76" x="6782" y="0"></use><use xlink:href="#E205-MJMAIN-65" x="7310" y="0"></use><use xlink:href="#E205-MJMAIN-72" x="7754" y="0"></use><use xlink:href="#E205-MJMAIN-67" x="8146" y="0"></use><use xlink:href="#E205-MJMAIN-65" x="8646" y="0"></use><use xlink:href="#E205-MJMAIN-6E" x="9090" y="0"></use><use xlink:href="#E205-MJMAIN-63" x="9646" y="0"></use><use xlink:href="#E205-MJMAIN-65" x="10090" y="0"></use><use xlink:href="#E205-MJMAIN-3A" x="10534" y="0"></use><use xlink:href="#E205-MJMAIN-7B" x="11089" y="0"></use></g><g transform="translate(0,44)"><use xlink:href="#E205-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E205-MJMATHI-6A" x="663" y="-213"></use><g transform="translate(1138,0)"><use xlink:href="#E205-MJMAIN-3A"></use><use xlink:href="#E205-MJMAIN-3D" x="278" y="0"></use></g><g transform="translate(2471,0)"><use xlink:href="#E205-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E205-MJMATHI-6A" x="663" y="-213"></use></g><use xlink:href="#E205-MJMAIN-2212" x="3554" y="0"></use><use xlink:href="#E205-MJMATHI-3B1" x="4554" y="0"></use><g transform="translate(5194,0)"><g transform="translate(120,0)"><rect stroke="none" width="1547" height="60" x="0" y="220"></rect><use xlink:href="#E205-MJMAIN-2202" x="490" y="676"></use><g transform="translate(60,-686)"><use xlink:href="#E205-MJMAIN-2202" x="0" y="0"></use><g transform="translate(567,0)"><use xlink:href="#E205-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E205-MJMATHI-6A" x="663" y="-213"></use></g></g></g></g><use xlink:href="#E205-MJMATHI-4A" x="6981" y="0"></use><g transform="translate(7781,0)"><use xlink:href="#E205-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E205-MJMATHI-3B8" x="389" y="0"></use><use xlink:href="#E205-MJMAIN-29" x="858" y="0"></use></g></g></g></g></g></svg></span><script type="math/tex">\begin{align*} & \text{repeat until convergence:} \; \lbrace \newline \; &{{\theta }_{j}}:={{\theta }_{j}}-\alpha \frac{\partial }{\partial {{\theta }_{j}}}J\left( {\theta}  \right) \newline \rbrace \end{align*}</script></p><p>解出偏导得：</p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="60.276ex" height="13.083ex" viewBox="0 -3067.8 25952.1 5633" role="img" focusable="false" style="vertical-align: -5.958ex;"><defs><path stroke-width="0" id="E105-MJMAIN-72" d="M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441T171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 373Q364 352 351 337T313 322Q288 322 276 338T263 372Q263 381 265 388T270 400T273 405Q271 407 250 401Q234 393 226 386Q179 341 179 207V154Q179 141 179 127T179 101T180 81T180 66V61Q181 59 183 57T188 54T193 51T200 49T207 48T216 47T225 47T235 46T245 46H276V0H267Q249 3 140 3Q37 3 28 0H20V46H36Z"></path><path stroke-width="0" id="E105-MJMAIN-65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z"></path><path stroke-width="0" id="E105-MJMAIN-70" d="M36 -148H50Q89 -148 97 -134V-126Q97 -119 97 -107T97 -77T98 -38T98 6T98 55T98 106Q98 140 98 177T98 243T98 296T97 335T97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 61 434T98 436Q115 437 135 438T165 441T176 442H179V416L180 390L188 397Q247 441 326 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y="0"></use></g></g></g></g></svg></span><script type="math/tex">\begin{align*}& \text{repeat until convergence:} \; \lbrace \newline \; & \theta_j := \theta_j - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_j^{(i)} \; & \text{for j := 0,1...n}\newline \rbrace\end{align*}</script></p><p>注意，虽然形式上梯度下降算法同线性回归一样，但其中的假设函不同，即<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.009ex" height="3.044ex" viewBox="0 -906.7 6892.7 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="0" id="E206-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="0" id="E206-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E206-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="0" id="E206-MJMATHI-78" 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stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E206-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E206-MJMATHI-3B8" x="814" y="-218"></use><use xlink:href="#E206-MJMAIN-28" x="1007" y="0"></use><use xlink:href="#E206-MJMATHI-78" x="1396" y="0"></use><use xlink:href="#E206-MJMAIN-29" x="1968" y="0"></use><use xlink:href="#E206-MJMAIN-3D" x="2635" y="0"></use><use xlink:href="#E206-MJMATHI-67" x="3691" y="0"></use><g transform="translate(4337,0)"><use xlink:href="#E206-MJSZ1-28"></use><g transform="translate(458,0)"><use xlink:href="#E206-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E206-MJMATHI-54" x="663" y="513"></use></g><use xlink:href="#E206-MJMATHI-78" x="1524" y="0"></use><use xlink:href="#E206-MJSZ1-29" x="2096" y="-1"></use></g></g></svg></span><script type="math/tex">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 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682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="0" id="E207-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 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type="math/tex">=-{{y}^{(i)}}\log \left( 1+{{e}^{-z}} \right)-\left( 1-{{y}^{(i)}} \right)\log \left( 1+{{e}^{z}} \right)</script></p><p>忆及 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.067ex" height="2.461ex" viewBox="0 -956.9 4334.4 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E214-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="0" id="E214-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="0" id="E214-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="0" id="E214-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="0" id="E214-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="0" id="E214-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="0" id="E214-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="0" id="E214-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 xlink:href="#E214-MJMATHI-7A" x="0" y="0"></use><use xlink:href="#E214-MJMAIN-3D" x="745" y="0"></use><g transform="translate(1801,0)"><use xlink:href="#E214-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E214-MJMATHI-54" x="663" y="513"></use></g><g transform="translate(2868,0)"><use xlink:href="#E214-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E214-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E214-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E214-MJMAIN-29" x="733" y="0"></use></g></g></g></svg></span><script type="math/tex">z=\theta^Tx^{(i)}</script>，对 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.998ex" height="2.694ex" viewBox="0 -806.1 860.3 1160" role="img" focusable="false" style="vertical-align: -0.822ex;"><defs><path stroke-width="0" id="E216-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="0" id="E216-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 xlink:href="#E216-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E216-MJMATHI-6A" x="663" y="-213"></use></g></svg></span><script type="math/tex">\theta_j</script> 求偏导则没有 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.998ex" height="2.694ex" viewBox="0 -806.1 860.3 1160" role="img" focusable="false" style="vertical-align: -0.822ex;"><defs><path stroke-width="0" id="E216-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="0" id="E216-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 xlink:href="#E216-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E216-MJMATHI-6A" x="663" y="-213"></use></g></svg></span><script type="math/tex">\theta_j</script> 的项求偏导即为 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.161ex" height="1.994ex" viewBox="0 -755.9 500 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E45-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E45-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex">0</script>，都消去，则得：</p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="23.971ex" height="4.212ex" viewBox="0 -1107.7 10320.6 1813.3" role="img" focusable="false" style="vertical-align: -1.639ex;"><defs><path stroke-width="0" id="E217-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="0" id="E217-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="0" id="E217-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E217-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="0" id="E217-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E217-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="0" id="E217-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="0" id="E217-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="0" id="E217-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="0" id="E217-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="0" id="E217-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="0" id="E217-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)"><g transform="translate(120,0)"><rect stroke="none" width="1129" height="60" x="0" y="220"></rect><g transform="translate(198,419)"><use transform="scale(0.707)" xlink:href="#E217-MJMAIN-2202" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E217-MJMATHI-7A" x="567" y="0"></use></g><g transform="translate(60,-410)"><use transform="scale(0.707)" xlink:href="#E217-MJMAIN-2202" x="0" y="0"></use><g transform="translate(400,0)"><use transform="scale(0.707)" xlink:href="#E217-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E217-MJMATHI-6A" x="663" y="-213"></use></g></g></g><use xlink:href="#E217-MJMAIN-3D" x="1647" y="0"></use><g transform="translate(2425,0)"><g transform="translate(397,0)"><rect stroke="none" width="1129" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E217-MJMAIN-2202" x="515" y="593"></use><g transform="translate(60,-410)"><use transform="scale(0.707)" xlink:href="#E217-MJMAIN-2202" x="0" y="0"></use><g transform="translate(400,0)"><use transform="scale(0.707)" xlink:href="#E217-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E217-MJMATHI-6A" x="663" y="-213"></use></g></g></g></g><g transform="translate(4072,0)"><use xlink:href="#E217-MJSZ1-28"></use><g transform="translate(458,0)"><use xlink:href="#E217-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E217-MJMATHI-54" x="663" y="513"></use></g><g transform="translate(1524,0)"><use xlink:href="#E217-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E217-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E217-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E217-MJMAIN-29" x="733" y="0"></use></g></g><use xlink:href="#E217-MJSZ1-29" x="2990" y="-1"></use></g><use xlink:href="#E217-MJMAIN-3D" x="7798" y="0"></use><g transform="translate(8854,0)"><use xlink:href="#E217-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,521)"><use transform="scale(0.707)" xlink:href="#E217-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E217-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E217-MJMAIN-29" x="733" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E217-MJMATHI-6A" x="808" y="-429"></use></g></g></svg></span><script type="math/tex">\frac{\partial z}{\partial {\theta_{j}}}=\frac{\partial }{\partial {\theta_{j}}}\left( \theta^Tx^{(i)}  \right)=x^{(i)}_j</script></p><p>所以有：</p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="56.326ex" height="3.978ex" viewBox="0 -1007.2 24251.3 1712.8" role="img" focusable="false" 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<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="24.514ex" height="3.745ex" viewBox="0 -1107.7 10554.6 1612.3" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="0" id="E226-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="0" id="E226-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E226-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="0" id="E226-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 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transform="translate(572,521)"><use transform="scale(0.707)" xlink:href="#E226-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E226-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E226-MJMAIN-29" x="733" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E226-MJMATHI-6A" x="808" y="-429"></use></g></g></g></svg></span><script type="math/tex">=-{\left({h_\theta}\left( {{x}^{(i)}} \right)-{{y}^{(i)}}\right)x_j^{(i)}}</script></p><p>则可得代价函数的导数：</p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="54.468ex" height="5.846ex" viewBox="0 -1459.5 23451.5 2517" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="0" id="E227-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 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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="0" id="E230-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 xlink:href="#E230-MJMATHI-4A" x="0" y="0"></use><use xlink:href="#E230-MJMAIN-28" x="633" y="0"></use><use xlink:href="#E230-MJMATHI-3B8" x="1022" y="0"></use><use xlink:href="#E230-MJMAIN-29" x="1491" y="0"></use></g></svg></span><script type="math/tex">J(\theta)</script> 值。</p><p>我们编写代码给出代价函数及其偏导数然后传入梯度下降算法中，接下来算法则会为我们最小化代价函数给出参数的最优解。这类算法被称为<strong>最优化算法(Optimization Algorithms)</strong>，梯度下降算法不是唯一的最小化算法<sup class='md-footnote'><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_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.486ex" height="1.41ex" viewBox="0 -504.6 640 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E72-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E72-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex">\alpha</script>（少一个参数少一份痛苦啊！）。</p><p>Octave/Matlab 中对这类高级算法做了封装，易于调用。</p><p>&nbsp;</p><p>假设有 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="28.31ex" height="2.811ex" viewBox="0 -906.7 12189.1 1210.2" role="img" focusable="false" style="vertical-align: 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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="0" id="E231-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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E231-MJMATHI-4A" x="0" y="0"></use><use xlink:href="#E231-MJMAIN-28" x="633" y="0"></use><use xlink:href="#E231-MJMATHI-3B8" x="1022" y="0"></use><use xlink:href="#E231-MJMAIN-29" x="1491" y="0"></use><use xlink:href="#E231-MJMAIN-3D" x="2157" y="0"></use><use xlink:href="#E231-MJMAIN-28" x="3213" y="0"></use><g transform="translate(3602,0)"><use xlink:href="#E231-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E231-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E231-MJMAIN-2212" x="4747" y="0"></use><use xlink:href="#E231-MJMAIN-35" x="5747" y="0"></use><g transform="translate(6247,0)"><use xlink:href="#E231-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E231-MJMAIN-32" x="550" y="513"></use></g><use xlink:href="#E231-MJMAIN-2B" x="7312" y="0"></use><use xlink:href="#E231-MJMAIN-28" x="8312" y="0"></use><g transform="translate(8701,0)"><use xlink:href="#E231-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E231-MJMAIN-32" x="663" y="-213"></use></g><use xlink:href="#E231-MJMAIN-2212" x="9846" y="0"></use><use xlink:href="#E231-MJMAIN-35" x="10846" y="0"></use><g transform="translate(11346,0)"><use xlink:href="#E231-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E231-MJMAIN-32" x="550" y="513"></use></g></g></svg></span><script type="math/tex">J(\theta) = (\theta_1-5)^2 + (\theta_2-5)^2</script>，要求参数 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.523ex" height="5.846ex" viewBox="0 -1509.8 4100.2 2517" role="img" focusable="false" style="vertical-align: -2.339ex;"><defs><path stroke-width="0" id="E232-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="0" id="E232-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="0" id="E232-MJMAIN-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path stroke-width="0" id="E232-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="0" id="E232-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="0" id="E232-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path stroke-width="0" id="E232-MJSZ3-5B" d="M247 -949V1450H516V1388H309V-887H516V-949H247Z"></path><path stroke-width="0" id="E232-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 xlink:href="#E232-MJMATHI-3B8" x="0" y="0"></use><use xlink:href="#E232-MJMAIN-3D" x="746" y="0"></use><g transform="translate(1802,0)"><use xlink:href="#E232-MJSZ3-5B"></use><g transform="translate(695,0)"><g transform="translate(-15,0)"><g transform="translate(0,650)"><use xlink:href="#E232-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E232-MJMAIN-31" x="663" y="-213"></use></g><g transform="translate(0,-750)"><use xlink:href="#E232-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E232-MJMAIN-32" x="663" y="-213"></use></g></g></g><use xlink:href="#E232-MJSZ3-5D" x="1769" y="-1"></use></g></g></svg></span><script type="math/tex">\theta=\begin{bmatrix} \theta_1\\\theta_2\end{bmatrix}</script>的最优值。</p><p>下面为 Octave/Matlab 求解最优化问题的代码实例：</p><ol start='' ><li>创建一个函数以返回代价函数及其偏导数：</li></ol><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="matlab"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></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: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>10</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><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: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">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: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">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: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">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: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">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: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">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: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">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: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">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: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">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: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">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: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">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: 0px; width: 1px; border-bottom: 0px solid transparent; top: 230px;"></div><div class="CodeMirror-gutters" style="height: 230px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></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 spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="matlab"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 36px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></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: 28px; margin-bottom: 0px; border-right-width: 0px; 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: -28px; width: 28px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 19px;">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: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 19px;">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: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 19px;">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: 0px; width: 1px; border-bottom: 0px solid transparent; top: 69px;"></div><div class="CodeMirror-gutters" style="height: 69px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 27px;"></div></div></div></div></pre><blockquote><p><code>&#39;GradObj&#39;, &#39;on&#39;</code>: 启用梯度目标参数（则需要将梯度传入算法）</p><p><code>&#39;MaxIter&#39;, 100</code>: 最大迭代次数为 100 次</p><p><code>@xxx</code>: Octave/Matlab 中的函数指针</p><p><code>optTheta</code>: 最优化得到的参数向量</p><p><code>functionVal</code>: 引用函数最后一次的返回值</p><p><code>exitFlag</code>: 标记代价函数是否收敛</p></blockquote><p>注：Octave/Matlab 中可以使用 <code>help fminunc</code> 命令随时查看函数的帮助文档。</p><ol start='3' ><li>返回结果</li></ol><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang=""><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 36px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></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: 28px; margin-bottom: 0px; border-right-width: 0px; 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" style=""><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: -28px; width: 28px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 19px;">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: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 19px;">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: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 19px;">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: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 19px;">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: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 19px;">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: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 19px;">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: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 19px;">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: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 19px;">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: 0px; width: 1px; border-bottom: 0px solid transparent; top: 184px;"></div><div class="CodeMirror-gutters" style="height: 184px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 27px;"></div></div></div></div></pre><h2><a name='header-n148' class='md-header-anchor '></a>6.7 多类别分类: 一对多(Multiclass Classification: One-vs-all)</h2><p>一直在讨论二元分类问题，这里谈谈多类别分类问题（比如天气预报）。</p><p><img src='images/20180112_001720.png' alt='' referrerPolicy='no-referrer' /></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_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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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="0" id="E234-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="0" id="E234-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 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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="0" id="E235-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="0" id="E235-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 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role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E22-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 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 xlink:href="#E22-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex">i</script> 个分类）的可能性</p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.21ex" height="1.994ex" viewBox="0 -755.9 521 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E242-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" 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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="0" id="E237-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="0" id="E237-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 xlink:href="#E237-MJMATHI-6B" x="0" y="0"></use><use xlink:href="#E237-MJMAIN-3D" x="798" y="0"></use><use xlink:href="#E237-MJMAIN-33" x="1854" y="0"></use></g></svg></span><script type="math/tex">k=3</script>。</p></blockquote><p>注意多类别分类问题中 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.476ex" height="2.577ex" viewBox="0 -806.1 2357.6 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E243-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 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y="0"></use></g></svg></span><script type="math/tex">h_\theta(x)</script> 就是一个 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.21ex" height="1.994ex" viewBox="0 -755.9 521 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E242-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 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x="1143" y="-913"></use></g><g transform="translate(4024,0)"><use xlink:href="#E245-MJMATHI-68" x="0" y="0"></use><g transform="translate(576,521)"><use transform="scale(0.707)" xlink:href="#E245-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E245-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E245-MJMAIN-29" x="733" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E245-MJMATHI-3B8" x="814" y="-473"></use></g><g transform="translate(5661,0)"><use xlink:href="#E245-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E245-MJMATHI-78" x="389" y="0"></use><use xlink:href="#E245-MJMAIN-29" x="961" y="0"></use></g></g></svg></span><script type="math/tex">y = \mathop{\max}\limits_i\,h_\theta^{\left( i \right)}\left( x \right)</script>。</p><h1><a name='header-n158' class='md-header-anchor '></a>7 正则化(Regularization)</h1><h2><a name='header-n159' class='md-header-anchor '></a>7.1 过拟合问题(The Problem of Overfitting)</h2><p>对于拟合的表现，可以分为三类情况：</p><ul><li><p><strong>欠拟合(Underfitting)</strong></p><p>无法很好的拟合训练集中的数据，预测值和实际值的误差很大，这类情况被称为欠拟合。拟合模型比较简单（特征选少了）时易出现这类情况。类似于，你上课不好好听，啥都不会，下课也差不多啥都不会。</p></li><li><p><strong>优良的拟合(Just right)</strong></p><p>不论是训练集数据还是不在训练集中的预测数据，都能给出较为正确的结果。类似于，学霸学神！</p></li><li><p><strong>过拟合(Overfitting)</strong></p><p>能很好甚至完美拟合训练集中的数据，即 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.141ex" height="2.577ex" viewBox="0 -806.1 3935.6 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E246-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 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0 0)"><use xlink:href="#E246-MJMATHI-4A" x="0" y="0"></use><use xlink:href="#E246-MJMAIN-28" x="633" y="0"></use><use xlink:href="#E246-MJMATHI-3B8" x="1022" y="0"></use><use xlink:href="#E246-MJMAIN-29" x="1491" y="0"></use><use xlink:href="#E246-MJMAIN-2192" x="2157" y="0"></use><use xlink:href="#E246-MJMAIN-30" x="3435" y="0"></use></g></svg></span><script type="math/tex">J(\theta) \to 0</script>，但是对于不在训练集中的<strong>新数据</strong>，预测值和实际值的误差会很大，<strong>泛化能力弱</strong>，这类情况被称为过拟合。拟合模型过于复杂（特征选多了）时易出现这类情况。类似于，你上课跟着老师做题都会都听懂了，下课遇到新题就懵了不会拓展。</p></li></ul><p>线性模型中的拟合情况(左图欠拟合，右图过拟合)：

<img src='images/20180112_091654.png' alt='' referrerPolicy='no-referrer' /></p><p>逻辑分类模型中的拟合情况：

<img src='images/20180112_092027.png' alt='' referrerPolicy='no-referrer' /></p><p>&nbsp;</p><p>为了度量拟合表现，引入：</p><ul><li><p>偏差(bias)</p><p>指模型的预测值与真实值的<strong>偏离程度</strong>。偏差越大，预测值偏离真实值越厉害。偏差低意味着能较好地反应训练集中的数据情况。</p></li><li><p>方差(Variance)</p><p>指模型预测值的<strong>离散程度或者变化范围</strong>。方差越大，数据的分布越分散，函数波动越大，泛化能力越差。方差低意味着拟合曲线的稳定性高，波动小。</p></li></ul><p>据此，我们有对同一数据的各类拟合情况如下图：

<img src='images/20180112_085630.png' alt='' referrerPolicy='no-referrer' /></p><p>据上图，高偏差意味着欠拟合，高方差意味着过拟合。</p><p>我们应尽量使得拟合模型处于低方差（较好地拟合数据）状态且同时处于低偏差（较好地预测新值）的状态。</p><p>避免过拟合的方法有：</p><ul><li><p>减少特征的数量</p><ul><li>手动选取需保留的特征</li><li>使用模型选择算法来选取合适的特征(如 PCA 算法)</li><li>减少特征的方式易丢失有用的特征信息</li></ul></li><li><p>正则化(Regularization)</p><ul><li>可保留所有参数（许多有用的特征都能轻微影响结果）</li><li>减少/惩罚各参数大小(magnitude)，以减轻各参数对模型的影响程度</li><li>当有很多参数对于模型只有轻微影响时，正则化方法的表现很好</li></ul></li></ul><h2><a name='header-n205' class='md-header-anchor '></a>7.2 代价函数(Cost Function)</h2><p>很多时候由于特征数量过多，过拟合时我们很难选出要保留的特征，这时候应用正则化方法则是很好的选择。</p><p>上文中，<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="30.545ex" height="2.694ex" viewBox="0 -906.7 13151.2 1160" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E247-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 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x="663" y="-213"></use></g><g transform="translate(12125,0)"><use xlink:href="#E247-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E247-MJMAIN-34" x="808" y="513"></use></g></g></svg></span><script type="math/tex">\theta_0 + \theta_1x + \theta_2x^2 + \theta_3x^3 + \theta_4x^4</script> 这样一个复杂的多项式较易过拟合，在不减少特征的情况下，<strong>如果能消除类似于 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.525ex" height="2.694ex" viewBox="0 -906.7 1948.1 1160" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E256-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 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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 xlink:href="#E256-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E256-MJMAIN-33" x="663" y="-213"></use><g transform="translate(922,0)"><use xlink:href="#E256-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E256-MJMAIN-33" x="808" y="513"></use></g></g></svg></span><script type="math/tex">\theta_3x^3</script>、<span class="MathJax_SVG" tabindex="-1" 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(h_\theta(x^{(i)}) - y^{(i)})^2 + 1000\cdot\theta_3^2 + 1000\cdot\theta_4^2</script></p><p>上式中，我们在代价函数中增加了 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.143ex" height="2.461ex" viewBox="0 -806.1 922.6 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E254-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="0" id="E254-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 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21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="0" id="E256-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E256-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E256-MJMAIN-33" x="663" y="-213"></use><g transform="translate(922,0)"><use xlink:href="#E256-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E256-MJMAIN-33" x="808" y="513"></use></g></g></svg></span><script type="math/tex">\theta_3x^3</script>、<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.525ex" height="2.577ex" viewBox="0 -906.7 1948.1 1109.7" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E257-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="0" id="E257-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path stroke-width="0" id="E257-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E257-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E257-MJMAIN-34" x="663" y="-213"></use><g transform="translate(922,0)"><use xlink:href="#E257-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E257-MJMAIN-34" x="808" y="513"></use></g></g></svg></span><script type="math/tex">\theta_4x^4</script> 这两项的参数非常小，就相当于没有了，假设函数也就<strong>“变得”简单</strong>了，从而在保留各参数的情况下避免了过拟合问题。</p><p><img src='images/20180114_090054.png' alt='' referrerPolicy='no-referrer' /></p><p>&nbsp;</p><p>根据上面的讨论，有时也无法决定要减少哪个参数，故统一惩罚除了 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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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 xlink:href="#E264-MJMATHI-78" x="0" y="0"></use><use xlink:href="#E264-MJMAIN-3D" x="849" y="0"></use><g transform="translate(1905,0)"><use xlink:href="#E264-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E264-MJMAIN-30" x="663" y="-213"></use></g></g></svg></span><script type="math/tex">x = \theta_0</script> 的直线 )</li><li>无法正常去过拟问题</li><li>梯度下降可能无法收敛</li></ul></li><li><p>过小</p><ul><li>无法避免过拟合（等于没有）</li></ul></li></ul><blockquote><p>正则化符合奥卡姆剃刀(Occam&#39;s razor)原理。在所有可能选择的模型中，能够很好地解释已知数据并且十分简单才是最好的模型，也就是应该选择的模型。从贝叶斯估计的角度来看，正则化项对应于模型的先验概率。可以假设复杂的模型有较大的先验概率，简单的模型有较小的先验概率。</p></blockquote><blockquote><p>正则化是结构风险最小化策略的实现，是去过拟合问题的典型方法，虽然看起来多了个一参数多了一重麻烦，后文会介绍自动选取正则化参数的方法。模型越复杂，正则化参数值就越大。比如，正则化项可以是模型参数向量的范数。</p></blockquote><h2><a name='header-n241' class='md-header-anchor '></a>7.3 线性回归正则化(Regularized Linear Regression)</h2><p>应用正则化的线性回归梯度下降算法：</p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="72.2ex" height="20.788ex" viewBox="0 -4726.4 31085.9 8950.1" role="img" focusable="false" style="vertical-align: -9.81ex;"><defs><path stroke-width="0" id="E274-MJMAIN-52" d="M130 622Q123 629 119 631T103 634T60 637H27V683H202H236H300Q376 683 417 677T500 648Q595 600 609 517Q610 512 610 501Q610 468 594 439T556 392T511 361T472 343L456 338Q459 335 467 332Q497 316 516 298T545 254T559 211T568 155T578 94Q588 46 602 31T640 16H645Q660 16 674 32T692 87Q692 98 696 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transform="scale(0.707)" xlink:href="#E266-MJMATHI-6A" x="808" y="-429"></use></g></g></svg></span><script type="math/tex">\theta_j := \theta_j(1 - \alpha\frac{\lambda}{m}) - \alpha\frac{1}{m}\sum_{i=1}^m(h_\theta(x^{(i)}) - y^{(i)})x_j^{(i)}</script></p><blockquote><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.276ex" height="3.278ex" viewBox="0 -956.9 1841.2 1411.3" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="0" id="E267-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E267-MJMATHI-6D" d="M21 287Q22 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transform="translate(2985,0)"><use xlink:href="#E268-MJMAIN-31" x="0" y="310"></use></g><g transform="translate(4485,0)"><use xlink:href="#E268-MJMAIN-22F1" x="0" y="-1810"></use></g><g transform="translate(6767,0)"><use xlink:href="#E268-MJMAIN-31" x="0" y="-3210"></use></g></g><g transform="translate(8268,3910)"><use xlink:href="#E268-MJSZ4-23A4" x="0" y="-1154"></use><g transform="translate(0,-5571) scale(1,6.348837209302325)"><use xlink:href="#E268-MJSZ4-23A5"></use></g><use xlink:href="#E268-MJSZ4-23A6" x="0" y="-6676"></use></g></g></g></g></g></g></svg></span><script type="math/tex">\begin{align*}& \theta = \left( X^TX + \lambda \cdot L \right)^{-1} X^Ty \newline& \text{where}\ \ L = \begin{bmatrix} 0 & & & & \newline & 1 & & & \newline & & 1 & & \newline & & & \ddots & \newline & & & & 1 \newline\end{bmatrix}\end{align*}</script></p><blockquote><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.614ex" height="1.994ex" viewBox="0 -755.9 1986.4 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E269-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E269-MJMAIN-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path stroke-width="0" id="E269-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E269-MJMATHI-3BB" x="0" y="0"></use><use xlink:href="#E269-MJMAIN-22C5" x="805" y="0"></use><use xlink:href="#E269-MJMATHI-4C" x="1305" y="0"></use></g></svg></span><script type="math/tex">\lambda\cdot L</script>: 正则化项</p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.582ex" height="1.994ex" viewBox="0 -755.9 681 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E270-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E270-MJMATHI-4C" x="0" y="0"></use></g></svg></span><script type="math/tex">L</script>: 第一行第一列为 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.161ex" height="1.994ex" viewBox="0 -755.9 500 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E45-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E45-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex">0</script> 的 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.394ex" height="2.11ex" viewBox="0 -755.9 2322.4 908.7" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="0" id="E99-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E99-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="0" id="E99-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 xlink:href="#E99-MJMATHI-6E" x="0" y="0"></use><use xlink:href="#E99-MJMAIN-2B" x="822" y="0"></use><use xlink:href="#E99-MJMAIN-31" x="1822" y="0"></use></g></svg></span><script type="math/tex">n+1</script> 维单位矩阵</p></blockquote><p>Matlab/Octave 代码：</p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang=""><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></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: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>10</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><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: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">&gt;&gt; L = eye(5)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">&gt;&gt; L(1,1) = 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">L =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">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: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 1 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 1 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 1 &nbsp; &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 0 &nbsp; &nbsp; 1</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 230px;"></div><div class="CodeMirror-gutters" style="height: 230px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p>&nbsp;</p><p>前文提到正则化可以解决正规方程法中不可逆的问题，即增加了 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.614ex" height="1.994ex" viewBox="0 -755.9 1986.4 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E271-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E271-MJMAIN-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path stroke-width="0" id="E271-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 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47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="0" id="E272-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="0" id="E272-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="0" id="E272-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E272-MJMAIN-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path stroke-width="0" id="E272-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E272-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E272-MJMATHI-54" x="1215" y="513"></use><use xlink:href="#E272-MJMATHI-58" x="1457" y="0"></use><use xlink:href="#E272-MJMAIN-2B" x="2531" y="0"></use><use xlink:href="#E272-MJMATHI-3BB" x="3531" y="0"></use><use xlink:href="#E272-MJMAIN-22C5" x="4336" y="0"></use><use xlink:href="#E272-MJMATHI-4C" x="4836" y="0"></use></g></svg></span><script type="math/tex">X^TX + \lambda \cdot L</script> 可逆(invertible)，即便 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.363ex" height="2.227ex" viewBox="0 -906.7 2309.1 958.9" role="img" focusable="false" 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y="0"></use><g transform="translate(572,521)"><use transform="scale(0.707)" xlink:href="#E274-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E274-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E274-MJMAIN-29" x="733" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E274-MJMATHI-6A" x="808" y="-429"></use></g><use xlink:href="#E274-MJSZ4-29" x="12002" y="0"></use></g><use xlink:href="#E274-MJMAIN-2B" x="13599" y="0"></use><g transform="translate(14377,0)"><g transform="translate(342,0)"><rect stroke="none" width="998" height="60" x="0" y="220"></rect><use xlink:href="#E274-MJMATHI-3BB" x="207" y="676"></use><use xlink:href="#E274-MJMATHI-6D" x="60" y="-686"></use></g></g><g transform="translate(15837,0)"><use xlink:href="#E274-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E274-MJMATHI-6A" x="663" y="-213"></use></g><use xlink:href="#E274-MJSZ4-5D" x="16697" y="-1"></use></g><use xlink:href="#E274-MJMAIN-2C" x="24058" y="0"></use><use xlink:href="#E274-MJMATHI-6A" x="25253" y="0"></use><use xlink:href="#E274-MJMAIN-2208" x="25943" y="0"></use><use xlink:href="#E274-MJMAIN-7B" x="26888" y="0"></use><use xlink:href="#E274-MJMAIN-31" x="27388" y="0"></use><use xlink:href="#E274-MJMAIN-2C" x="27888" y="0"></use><g transform="translate(28332,0)"><use xlink:href="#E274-MJMAIN-32"></use><use xlink:href="#E274-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E274-MJMAIN-2E" x="778" y="0"></use><use xlink:href="#E274-MJMAIN-2E" x="1056" y="0"></use></g><use xlink:href="#E274-MJMATHI-6E" x="29666" y="0"></use><use xlink:href="#E274-MJMAIN-7D" x="30266" y="0"></use></g><g transform="translate(0,-3880)"><use xlink:href="#E274-MJMAIN-7D" x="0" y="0"></use></g></g></g></g></svg></span><script type="math/tex">\begin{align*} & \text{Repeat}\ \lbrace \newline & \ \ \ \ \theta_0 := \theta_0 - \alpha\ \frac{1}{m}\ \sum_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)})x_0^{(i)} \newline & \ \ \ \ \theta_j := \theta_j - \alpha\ \left[ \left( \frac{1}{m}\ \sum_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)})x_j^{(i)} \right) + \frac{\lambda}{m}\theta_j \right], \ \ \ j \in \lbrace 1,2...n\rbrace\newline & \rbrace \end{align*}</script></p><p>&nbsp;</p><div class='footnotes-area'  ><hr/>

<div class='footnote-line'><span class='md-fn-count'>1</span> <a href='https://en.wikipedia.org/wiki/List_of_algorithms#Optimization_algorithms' target='_blank' class='url'>https://en.wikipedia.org/wiki/List_of_algorithms#Optimization_algorithms</a><a name='dfref-footnote-1' href='#ref-footnote-1' title='回到文档' class='reversefootnote' >↩</a></div>

<div class='footnote-line'><span class='md-fn-count'>2</span> week2 - 4.6<a name='dfref-footnote-2' href='#ref-footnote-2' title='回到文档' class='reversefootnote' >↩</a></div></div></div>

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