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  223. <div  id='write'  class = 'is-node show-fences-line-number'><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" href="#header-n2">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="n40"><a class="md-toc-inner" href="#header-n40">6.2 假设函数表示(Hypothesis Representation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n62"><a class="md-toc-inner" href="#header-n62">6.3 决策边界(Decision Boundary)</a></span><span class="md-toc-item md-toc-h2" data-ref="n65"><a class="md-toc-inner" href="#header-n65">6.4 Cost Function</a></span><span class="md-toc-item md-toc-h2" data-ref="n66"><a class="md-toc-inner" href="#header-n66">6.5 Simplified Cost Function and Gradient Descent</a></span><span class="md-toc-item md-toc-h2" data-ref="n67"><a class="md-toc-inner" href="#header-n67">6.6 Advanced Optimization</a></span><span class="md-toc-item md-toc-h2" data-ref="n68"><a class="md-toc-inner" href="#header-n68">6.7 Multiclass Classification_ One-vs-all</a></span><span class="md-toc-item md-toc-h1" data-ref="n69"><a class="md-toc-inner" href="#header-n69">7 Regularization</a></span><span class="md-toc-item md-toc-h2" data-ref="n70"><a class="md-toc-inner" href="#header-n70">7.1 The Problem of Overfitting</a></span><span class="md-toc-item md-toc-h2" data-ref="n71"><a class="md-toc-inner" href="#header-n71">7.2 Cost Function</a></span><span class="md-toc-item md-toc-h2" data-ref="n72"><a class="md-toc-inner" href="#header-n72">7.3 Regularized Linear Regression</a></span><span class="md-toc-item md-toc-h2" data-ref="n73"><a class="md-toc-inner" href="#header-n73">7.4 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>肿瘤诊断问题：</p><p><img src='image/20180109_144040.png' alt='' /></p><p>肿瘤诊断问题是一个<strong>二元分类问题(binary class problems)</strong>，则定义 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-86-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.68ex" height="2.577ex" viewBox="0 -806.1 4167.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E86-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 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表示<strong>负向类(negative class)</strong>，代表恶性肿瘤(&quot;-&quot;)，1 为<strong>正向类(positive class)</strong>，代表良性肿瘤(&quot;+&quot;)。如图，定义最右边的样本为<strong>偏差项</strong>。</p><p>在未加入偏差项时，线性回归算法给出了品红色的拟合直线，若规定</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-87-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.551ex" height="2.577ex" viewBox="0 -806.1 4973.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E88-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMAIN-3C" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-89">h_\theta(x) \lt 0.5</script> ，预测为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-90-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" 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0</script>，即负向类。</p><p>即以 0.5 为<strong>阈值</strong>(threshold)，则我们就可以根据线性回归结果，得到相对正确的分类结果 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-20-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.155ex" height="1.877ex" viewBox="0 -504.6 497.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E20-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E20-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-20">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。</p><p>不仅如此，线性回归算法的值域为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-91-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="1.994ex" viewBox="0 -755.9 759.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E92-MJMATHI-52" d="M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E92-MJMATHI-52" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-91">R</script>，则当线性回归函数给出诸如 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-92-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="23.342ex" height="2.344ex" viewBox="0 -755.9 10049.8 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E93-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 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y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMAIN-3D" x="2804" y="0"></use><g transform="translate(3860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMATHI-58" x="4928" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-96">h_\theta \left( x \right)=\theta^{T}X</script>，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-97-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.116ex" height="1.877ex" viewBox="0 -504.6 480.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E98-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E98-MJMATHI-67" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-97">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，则为线性回归函数。</p><p>一个常用的逻辑函数是 S 形函数，叫做 <a href='https://en.wikipedia.org/wiki/Sigmoid_function' target='_blank'>sigmoid 函数</a>（如下图），其公式为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-98-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.814ex" height="3.395ex" viewBox="0 -956.9 5517.2 1461.5" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E99-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 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440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E99-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E99-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" 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transform="translate(59,-376)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E99-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E99-MJMAIN-2B" x="500" y="0"></use><g transform="translate(904,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E99-MJMATHI-65" x="0" y="0"></use><g transform="translate(329,204)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E99-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E99-MJMATHI-7A" x="778" y="0"></use></g></g></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-98">g\left( z \right)=\frac{1}{1+{{e}^{-z}}}</script>。 </p><p><img src='image/2413fbec8ff9fa1f19aaf78265b8a33b_Logistic_function.png' alt='sigmoid function' /></p><p>应用 sigmoid 函数，则逻辑回归模型：<span 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x="1685" y="572"></use><g transform="translate(60,-608)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMAIN-2B" x="500" y="0"></use><g transform="translate(904,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMATHI-65" x="0" y="0"></use><g transform="translate(329,204)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMAIN-2212" x="0" y="0"></use><g transform="translate(389,0)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMATHI-54" x="469" y="319"></use></g><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E100-MJMATHI-78" x="2052" y="0"></use></g></g></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-99">h_{\theta}(x)=g(\theta^Tx) =\frac{1}{1+e^{-\theta^Tx}}</script></p><p>逻辑回归模型中，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-100-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.868ex" height="2.577ex" viewBox="0 -806.1 2526.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E101-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 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E101-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-100">h_\theta \left( x \right)</script> 的作用是，根据输入 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-13-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.33ex" height="1.41ex" viewBox="0 -504.6 572.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path 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type="math/tex" id="MathJax-Element-69">\theta</script>，计算得出”输出 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-101-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E102-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E102-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E102-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E102-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E102-MJMAIN-3D" x="775" 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x="3751" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-3B" x="4324" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMATHI-3B8" x="4769" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-29" x="5238" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-2B" x="5850" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMATHI-50" x="6851" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-28" x="7602" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMATHI-79" x="7992" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-3D" x="8767" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-31" x="9823" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-7C" x="10324" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMATHI-78" x="10602" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-3B" x="11175" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMATHI-3B8" x="11620" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-29" x="12089" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-3D" x="12757" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-31" x="13813" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-102">\begin{align*}& h_\theta(x) = P(y=1 | x ; \theta) = 1 - P(y=0 | x ; \theta) \newline & P(y = 0 | x;\theta) + P(y = 1 | x ; \theta) = 1\end{align*}</script></p><p>&nbsp;</p><p>以肿瘤诊断为例，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-103-Frame" tabindex="-1" style="font-size: 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130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E104-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E104-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 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xlink:href="#E104-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-3D" x="2804" y="0"></use><g transform="translate(3860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-37" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-103">h_\theta \left( x \right)=0.7</script> 表示病人有 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-104-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.261ex" height="2.227ex" viewBox="0 -806.1 1834.5 958.9" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="1" id="E105-MJMAIN-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path><path stroke-width="1" id="E105-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E105-MJMAIN-25" d="M465 605Q428 605 394 614T340 632T319 641Q332 608 332 548Q332 458 293 403T202 347Q145 347 101 402T56 548Q56 637 101 693T202 750Q241 750 272 719Q359 642 464 642Q580 642 650 732Q662 748 668 749Q670 750 673 750Q682 750 688 743T693 726Q178 -47 170 -52Q166 -56 160 -56Q147 -56 142 -45Q137 -36 142 -27Q143 -24 363 304Q469 462 525 546T581 630Q528 605 465 605ZM207 385Q235 385 263 427T292 548Q292 617 267 664T200 712Q193 712 186 709T167 698T147 668T134 615Q132 595 132 548V527Q132 436 165 403Q183 385 203 385H207ZM500 146Q500 234 544 290T647 347Q699 347 737 292T776 146T737 0T646 -56Q590 -56 545 0T500 146ZM651 -18Q679 -18 707 24T736 146Q736 215 711 262T644 309Q637 309 630 306T611 295T591 265T578 212Q577 200 577 146V124Q577 -18 647 -18H651Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E105-MJMAIN-37"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E105-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E105-MJMAIN-25" x="1001" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-104">70\%</script> 的概率得了恶性肿瘤。</p><h2><a name='header-n62' class='md-header-anchor '></a>6.3 决策边界(Decision Boundary)</h2><p>&nbsp;</p><h2><a name='header-n65' class='md-header-anchor '></a>6.4 Cost Function</h2><h2><a name='header-n66' class='md-header-anchor '></a>6.5 Simplified Cost Function and Gradient Descent</h2><h2><a name='header-n67' class='md-header-anchor '></a>6.6 Advanced Optimization</h2><h2><a name='header-n68' class='md-header-anchor '></a>6.7 Multiclass Classification_ One-vs-all</h2><h1><a name='header-n69' class='md-header-anchor '></a>7 Regularization</h1><h2><a name='header-n70' class='md-header-anchor '></a>7.1 The Problem of Overfitting</h2><h2><a name='header-n71' class='md-header-anchor '></a>7.2 Cost Function</h2><h2><a name='header-n72' class='md-header-anchor '></a>7.3 Regularized Linear Regression</h2><h2><a name='header-n73' class='md-header-anchor '></a>7.4 Regularized Logistic Regression</h2></div>

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