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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" style="cursor: pointer;" href="#header-n2">9 神经网络: 学习(Neural Networks: Learning)</a></span><span class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n3">9.1 代价函数(Cost Function)</a></span><span class="md-toc-item md-toc-h2" data-ref="n43"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n43">9.2 反向传播算法(Backpropagation Algorithm)</a></span><span class="md-toc-item md-toc-h2" data-ref="n44"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n44">9.3 直观理解反向传播(Backpropagation Intuition)</a></span><span class="md-toc-item md-toc-h2" data-ref="n45"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n45">9.4 实现注意点: 参数展开(Implementation Note: Unrolling Parameters)</a></span><span class="md-toc-item md-toc-h2" data-ref="n46"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n46">9.5 Gradient Checking</a></span><span class="md-toc-item md-toc-h2" data-ref="n47"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n47">9.6 Random Initialization</a></span><span class="md-toc-item md-toc-h2" data-ref="n48"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n48">9.7 Putting It Together</a></span><span class="md-toc-item md-toc-h2" data-ref="n49"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n49">9.8 自主驾驶(Autonomous Driving)</a></span></p></div><h1><a name='header-n2' class='md-header-anchor '></a>9 神经网络: 学习(Neural Networks: Learning)</h1><h2><a name='header-n3' class='md-header-anchor '></a>9.1 代价函数(Cost Function)</h2><p>对于神经网络的代价函数公式：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.693ex" height="1.994ex" viewBox="0 -755.9 298.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E148-MJMATHI-6C" d="M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z"></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="#E148-MJMATHI-6C" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-149">l</script> 层激活单元的数量（不包含偏置单元）</p><p><span 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y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E591-MJMAIN-2208" x="1918" y="0"></use><g transform="translate(2863,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E591-MJAMS-52" x="0" y="0"></use><g transform="translate(722,410)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E591-MJMAIN-28" x="0" y="0"></use><g transform="translate(275,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E591-MJMATHI-73" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E591-MJMATHI-6C" x="663" y="-213"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E591-MJMAIN-2B" x="1170" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E591-MJMAIN-31" x="1948" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" 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635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z"></path><path stroke-width="1" id="E634-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E634-MJMAIN-398" x="0" y="0"></use><g transform="translate(778,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E634-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E634-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E634-MJMAIN-29" x="688" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-513">\Theta^{(l)}</script> ，依次平方后求取其除了偏置参数部分的和值，并循环累加即得结果。</li></ul><blockquote><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-599-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.353ex" height="2.11ex" viewBox="0 -806.1 1443.7 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E739-MJAMS-52" d="M17 665Q17 672 28 683H221Q415 681 439 677Q461 673 481 667T516 654T544 639T566 623T584 607T597 592T607 578T614 565T618 554L621 548Q626 530 626 497Q626 447 613 419Q578 348 473 326L455 321Q462 310 473 292T517 226T578 141T637 72T686 35Q705 30 705 16Q705 7 693 -1H510Q503 6 404 159L306 310H268V183Q270 67 271 59Q274 42 291 38Q295 37 319 35Q344 35 353 28Q362 17 353 3L346 -1H28Q16 5 16 16Q16 35 55 35Q96 38 101 52Q106 60 106 341T101 632Q95 645 55 648Q17 648 17 665ZM241 35Q238 42 237 45T235 78T233 163T233 337V621L237 635L244 648H133Q136 641 137 638T139 603T141 517T141 341Q141 131 140 89T134 37Q133 36 133 35H241ZM457 496Q457 540 449 570T425 615T400 634T377 643Q374 643 339 648Q300 648 281 635Q271 628 270 610T268 481V346H284Q327 346 375 352Q421 364 439 392T457 496ZM492 537T492 496T488 427T478 389T469 371T464 361Q464 360 465 360Q469 360 497 370Q593 400 593 495Q593 592 477 630L457 637L461 626Q474 611 488 561Q492 537 492 496ZM464 243Q411 317 410 317Q404 317 401 315Q384 315 370 312H346L526 35H619L606 50Q553 109 464 243Z"></path><path stroke-width="1" id="E739-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E739-MJAMS-52" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E739-MJMATHI-6D" x="1021" y="581"></use></g></svg></span><script type="math/tex" id="MathJax-Element-599">\mathbb{R}^{m}</script>: 即 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-18-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.04ex" height="1.41ex" viewBox="0 -504.6 878.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E18-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E18-MJMATHI-6D" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-18">m</script> 维向量</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-576-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.618ex" height="2.227ex" viewBox="0 -856.4 2418.8 958.9" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E700-MJAMS-52" d="M17 665Q17 672 28 683H221Q415 681 439 677Q461 673 481 667T516 654T544 639T566 623T584 607T597 592T607 578T614 565T618 554L621 548Q626 530 626 497Q626 447 613 419Q578 348 473 326L455 321Q462 310 473 292T517 226T578 141T637 72T686 35Q705 30 705 16Q705 7 693 -1H510Q503 6 404 159L306 310H268V183Q270 67 271 59Q274 42 291 38Q295 37 319 35Q344 35 353 28Q362 17 353 3L346 -1H28Q16 5 16 16Q16 35 55 35Q96 38 101 52Q106 60 106 341T101 632Q95 645 55 648Q17 648 17 665ZM241 35Q238 42 237 45T235 78T233 163T233 337V621L237 635L244 648H133Q136 641 137 638T139 603T141 517T141 341Q141 131 140 89T134 37Q133 36 133 35H241ZM457 496Q457 540 449 570T425 615T400 634T377 643Q374 643 339 648Q300 648 281 635Q271 628 270 610T268 481V346H284Q327 346 375 352Q421 364 439 392T457 496ZM492 537T492 496T488 427T478 389T469 371T464 361Q464 360 465 360Q469 360 497 370Q593 400 593 495Q593 592 477 630L457 637L461 626Q474 611 488 561Q492 537 492 496ZM464 243Q411 317 410 317Q404 317 401 315Q384 315 370 312H346L526 35H619L606 50Q553 109 464 243Z"></path><path stroke-width="1" id="E700-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E700-MJMAIN-D7" d="M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path><path stroke-width="1" id="E700-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E700-MJAMS-52" x="0" y="0"></use><g transform="translate(722,410)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E700-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E700-MJMAIN-D7" x="878" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E700-MJMATHI-6E" x="1657" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-576">\mathbb{R}^{m\times n}</script>: 即 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-589-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.276ex" height="1.527ex" viewBox="0 -554.9 2701.9 657.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E718-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E718-MJMAIN-D7" d="M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path><path stroke-width="1" id="E718-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E718-MJMATHI-6D" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E718-MJMAIN-D7" x="1100" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E718-MJMATHI-6E" x="2101" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-589">m \times n</script> 维矩阵</p></blockquote><p>可见，神经网络背后的思想是和逻辑回归一样的。</p><p>&nbsp;</p><h2><a name='header-n43' class='md-header-anchor '></a>9.2 反向传播算法(Backpropagation Algorithm)</h2><h2><a name='header-n44' class='md-header-anchor '></a>9.3 直观理解反向传播(Backpropagation Intuition)</h2><h2><a name='header-n45' class='md-header-anchor '></a>9.4 实现注意点: 参数展开(Implementation Note: Unrolling Parameters)</h2><h2><a name='header-n46' class='md-header-anchor '></a>9.5 Gradient Checking</h2><h2><a name='header-n47' class='md-header-anchor '></a>9.6 Random Initialization</h2><h2><a name='header-n48' class='md-header-anchor '></a>9.7 Putting It Together</h2><h2><a name='header-n49' class='md-header-anchor '></a>9.8 自主驾驶(Autonomous Driving)</h2></div>

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