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strugglesharp / ML-AndrewNg-Notes

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week5.html 693.93 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">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" href="#header-n3">9.1 代价函数(Cost Function)</a></span><span class="md-toc-item md-toc-h2" data-ref="n36"><a class="md-toc-inner" href="#header-n36">9.2 反向传播算法(Backpropagation Algorithm)</a></span><span class="md-toc-item md-toc-h2" data-ref="n89"><a class="md-toc-inner" href="#header-n89">9.3 直观理解反向传播(Backpropagation Intuition)</a></span><span class="md-toc-item md-toc-h2" data-ref="n138"><a class="md-toc-inner" href="#header-n138">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="n145"><a class="md-toc-inner" href="#header-n145">9.5 梯度检验(Gradient Checking)</a></span><span class="md-toc-item md-toc-h2" data-ref="n157"><a class="md-toc-inner" href="#header-n157">9.6 随机初始化(Random Initialization)</a></span><span class="md-toc-item md-toc-h2" data-ref="n167"><a class="md-toc-inner" href="#header-n167">9.7 综合起来(Putting It Together)</a></span><span class="md-toc-item md-toc-h2" data-ref="n199"><a class="md-toc-inner" href="#header-n199">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>神经网络的分类问题有两种：</p><ul><li><p>二元分类问题(0/1分类)</p><p>只有一个输出单元 (<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.323ex" height="1.877ex" viewBox="0 -755.9 2722.6 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E332-MJMATHI-4B" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path><path stroke-width="0" id="E332-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="E332-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="#E332-MJMATHI-4B" x="0" y="0"></use><use xlink:href="#E332-MJMAIN-3D" x="1166" y="0"></use><use xlink:href="#E332-MJMAIN-31" x="2222" y="0"></use></g></svg></span><script type="math/tex">K=1</script>)</p></li><li><p>多元(<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.065ex" height="1.877ex" viewBox="0 -755.9 889 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E329-MJMATHI-4B" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E329-MJMATHI-4B" x="0" y="0"></use></g></svg></span><script type="math/tex">K</script>)分类问题</p><p>输出单元不止一个(<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.323ex" height="1.994ex" viewBox="0 -755.9 2722.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex; max-width: 5400px;"><defs><path stroke-width="0" id="E333-MJMATHI-4B" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path><path stroke-width="0" id="E333-MJMAIN-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path stroke-width="0" id="E333-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="#E333-MJMATHI-4B" x="0" y="0"></use><use xlink:href="#E333-MJMAIN-3E" x="1166" y="0"></use><use xlink:href="#E333-MJMAIN-31" x="2222" y="0"></use></g></svg></span><script type="math/tex">K\gt1</script>)</p></li></ul><p>神经网络的代价函数公式：</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="41.184ex" height="2.928ex" viewBox="0 -956.9 17731.8 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E334-MJMATHI-68" d="M137 683Q138 683 209 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x="5564" y="0"></use></g></svg></span><script type="math/tex">P(y=k | x ; \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="2.065ex" height="1.877ex" viewBox="0 -755.9 889 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E329-MJMATHI-4B" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E329-MJMATHI-4B" x="0" y="0"></use></g></svg></span><script type="math/tex">K</script>: 输出层的输出单元数量，即类数 - 1</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="3.236ex" height="3.511ex" viewBox="0 -1107.7 1393.2 1511.8" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="0" id="E341-MJMATHI-79" d="M21 287Q21 301 36 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367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E242-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script> 个分量值</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="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" 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xlink:href="#E345-MJMAIN-31" x="1076" y="0"></use></g></g></g></g></svg></span><script type="math/tex">\sum\limits_{l=1}^{L-1}\sum\limits_{i=1}^{s_l}\sum\limits_{j=1}^{s_{l+1}}</script> ，就是对每一层间的多维矩权重 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.806ex" height="2.461ex" viewBox="0 -956.9 1638.8 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E436-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="0" id="E436-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 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749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E436-MJMAIN-398" x="0" y="0"></use><g transform="translate(778,362)"><use transform="scale(0.707)" xlink:href="#E436-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E436-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E436-MJMAIN-29" x="687" y="0"></use></g></g></svg></span><script type="math/tex">\Theta^{(l)}</script> ，依次平方后求取其除了偏置权重部分的和值，并循环累加即得结果。</li></ul><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="3.351ex" height="2.11ex" viewBox="0 -806.1 1442.8 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E347-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="0" id="E347-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 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xlink:href="#E347-MJMATHI-6D" x="1021" y="579"></use></g></svg></span><script type="math/tex">\mathbb{R}^{m}</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="2.039ex" height="1.41ex" viewBox="0 -504.6 878 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" 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 xlink:href="#E18-MJMATHI-6D" x="0" y="0"></use></g></svg></span><script type="math/tex">m</script> 维向量</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="5.614ex" height="2.227ex" viewBox="0 -856.4 2417.2 958.9" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E348-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="0" id="E348-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 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y="0"></use><use transform="scale(0.707)" xlink:href="#E348-MJMAIN-D7" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E348-MJMATHI-6E" x="1655" y="0"></use></g></g></svg></span><script type="math/tex">\mathbb{R}^{m\times n}</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="6.272ex" height="1.527ex" viewBox="0 -554.9 2700.4 657.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E349-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 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name='header-n36' class='md-header-anchor '></a>9.2 反向传播算法(Backpropagation Algorithm)</h2><p>类似于回归模型中的梯度下降算法，为了求解神经网络最优化问题，我们也要计算 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.129ex" height="3.511ex" viewBox="0 -1007.2 3500.1 1511.8" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="0" id="E354-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" 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。</p><p>在神经网络中，代价函数看上去虽然不复杂，但要注意到其中 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.983ex" height="2.577ex" viewBox="0 -806.1 2576.1 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E326-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="E326-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 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-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="#E358-MJMAIN-394" x="0" y="0"></use><g transform="translate(833,362)"><use transform="scale(0.707)" xlink:href="#E358-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E358-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E358-MJMAIN-29" x="687" y="0"></use></g></g></svg></span><script type="math/tex">\Delta^{(l)}</script> 为一个全零矩阵。</p></li><li><p>接下来遍历所有样本实例，对于每一个样本实例，有下列步骤：</p><ol start='' ><li><p>运行前向传播算法，得到初始预测 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.938ex" height="2.928ex" viewBox="0 -956.9 5570.4 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" 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xlink:href="#E359-MJMATHI-4C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E359-MJMAIN-29" x="1069" y="0"></use></g><use xlink:href="#E359-MJMAIN-3D" x="1938" y="0"></use><g transform="translate(2994,0)"><use xlink:href="#E359-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E359-MJMAIN-398" x="814" y="-218"></use></g><use xlink:href="#E359-MJMAIN-28" x="4220" y="0"></use><use xlink:href="#E359-MJMATHI-78" x="4609" y="0"></use><use xlink:href="#E359-MJMAIN-29" x="5181" y="0"></use></g></svg></span><script type="math/tex">a^{(L)}=h_\Theta(x)</script> 。</p></li><li><p>运行反向传播算法，从输出层开始计算每一层预测的<strong>误差</strong>(error)，以此来求取偏导。</p><p><img src='images/20180120_105744.png' alt='' referrerPolicy='no-referrer' /></p><p>输出层的误差即为预测与训练集结果的之间的差值：<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.629ex" height="2.928ex" viewBox="0 -956.9 6298.5 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E360-MJMATHI-3B4" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z"></path><path stroke-width="0" id="E360-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="E360-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 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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="#E360-MJMATHI-3B4" x="0" y="0"></use><g transform="translate(453,362)"><use transform="scale(0.707)" xlink:href="#E360-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E360-MJMATHI-4C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E360-MJMAIN-29" x="1069" y="0"></use></g><use xlink:href="#E360-MJMAIN-3D" x="1862" y="0"></use><g transform="translate(2918,0)"><use xlink:href="#E360-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,362)"><use transform="scale(0.707)" xlink:href="#E360-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E360-MJMATHI-4C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E360-MJMAIN-29" x="1069" y="0"></use></g></g><use xlink:href="#E360-MJMAIN-2212" x="4801" y="0"></use><use xlink:href="#E360-MJMATHI-79" x="5801" y="0"></use></g></svg></span><script type="math/tex">\delta^{(L)} = a^{(L)} - y</script>，</p><p>对于隐藏层中每一层的误差，都通过上一层的误差来计算：</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="55.329ex" height="3.861ex" viewBox="0 -1107.7 23822.2 1662.6" role="img" focusable="false" style="vertical-align: -1.289ex;"><defs><path stroke-width="0" id="E361-MJMATHI-3B4" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 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-238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E431-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="E431-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="E431-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E431-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,362)"><use transform="scale(0.707)" xlink:href="#E431-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E431-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E431-MJMAIN-29" x="687" y="0"></use></g><use xlink:href="#E431-MJMAIN-3D" x="1667" y="0"></use><use xlink:href="#E431-MJMAIN-28" x="2723" y="0"></use><use xlink:href="#E431-MJMATHI-67" x="3112" y="0"></use><use xlink:href="#E431-MJMAIN-28" x="3592" y="0"></use><g transform="translate(3981,0)"><use xlink:href="#E431-MJMATHI-7A" x="0" y="0"></use><g transform="translate(468,362)"><use transform="scale(0.707)" xlink:href="#E431-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E431-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E431-MJMAIN-29" x="687" y="0"></use></g></g><use xlink:href="#E431-MJMAIN-29" x="5311" y="0"></use></g></svg></span><script type="math/tex">a^{(l)} = (g(z^{(l)})</script> 添加偏置单元 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.39ex" height="3.511ex" viewBox="0 -1107.7 3612.4 1511.8" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="0" id="E432-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E432-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="E432-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><path stroke-width="0" id="E432-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="E432-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="E432-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="E432-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="#E432-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,521)"><use transform="scale(0.707)" xlink:href="#E432-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E432-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E432-MJMAIN-29" x="687" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E432-MJMAIN-30" x="748" y="-434"></use><use xlink:href="#E432-MJMAIN-3D" x="1667" y="0"></use><use xlink:href="#E432-MJMAIN-31" x="2723" y="0"></use><use xlink:href="#E432-MJMAIN-29" x="3223" y="0"></use></g></svg></span><script type="math/tex">a^{(l)}_0 = 1)</script></p><p>解得 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="43.456ex" height="3.628ex" viewBox="0 -1007.2 18710.1 1562" role="img" focusable="false" style="vertical-align: -1.289ex;"><defs><path stroke-width="0" id="E368-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="E368-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="E368-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 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y="0"></use></g></g></g></g><use xlink:href="#E368-MJMATHI-67" x="1701" y="0"></use><use xlink:href="#E368-MJMAIN-28" x="2181" y="0"></use><g transform="translate(2570,0)"><use xlink:href="#E368-MJMATHI-7A" x="0" y="0"></use><g transform="translate(468,362)"><use transform="scale(0.707)" xlink:href="#E368-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E368-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E368-MJMAIN-29" x="687" y="0"></use></g></g><use xlink:href="#E368-MJMAIN-29" x="3900" y="0"></use><use xlink:href="#E368-MJMAIN-3D" x="4566" y="0"></use><g transform="translate(5622,0)"><use xlink:href="#E368-MJMATHI-67" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E368-MJMAIN-2032" x="680" y="513"></use></g><use xlink:href="#E368-MJMAIN-28" x="6398" y="0"></use><g transform="translate(6787,0)"><use xlink:href="#E368-MJMATHI-7A" x="0" y="0"></use><g transform="translate(468,362)"><use transform="scale(0.707)" xlink:href="#E368-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E368-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E368-MJMAIN-29" x="687" y="0"></use></g></g><use xlink:href="#E368-MJMAIN-29" x="8116" y="0"></use><use xlink:href="#E368-MJMAIN-3D" x="8783" y="0"></use><use xlink:href="#E368-MJMATHI-67" x="9839" y="0"></use><use xlink:href="#E368-MJMAIN-28" x="10319" y="0"></use><g transform="translate(10708,0)"><use xlink:href="#E368-MJMATHI-7A" x="0" y="0"></use><g transform="translate(468,362)"><use transform="scale(0.707)" xlink:href="#E368-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E368-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E368-MJMAIN-29" x="687" y="0"></use></g></g><use xlink:href="#E368-MJMAIN-29" x="12038" y="0"></use><use xlink:href="#E368-MJMAIN-2E" x="12427" y="0"></use><use xlink:href="#E368-MJMAIN-2217" x="12871" y="0"></use><use xlink:href="#E368-MJMAIN-28" x="13621" y="0"></use><use xlink:href="#E368-MJMAIN-31" x="14010" y="0"></use><use xlink:href="#E368-MJMAIN-2212" x="14733" y="0"></use><use xlink:href="#E368-MJMATHI-67" x="15733" y="0"></use><use xlink:href="#E368-MJMAIN-28" x="16213" y="0"></use><g transform="translate(16602,0)"><use xlink:href="#E368-MJMATHI-7A" x="0" y="0"></use><g transform="translate(468,362)"><use transform="scale(0.707)" xlink:href="#E368-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E368-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E368-MJMAIN-29" x="687" y="0"></use></g></g><use xlink:href="#E368-MJMAIN-29" x="17932" y="0"></use><use xlink:href="#E368-MJMAIN-29" x="18321" y="0"></use></g></svg></span><script type="math/tex">\frac{\partial}{\partial z^{(l)}}g(z^{(l)})=g'(z^{(l)})=g(z^{(l)}) .* \ (1-g(z^{(l)}))</script>，</p><p>则有 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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transform="scale(0.707)" xlink:href="#E369-MJMAIN-29" x="687" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E369-MJMAIN-30" x="748" y="-434"></use></g><use xlink:href="#E369-MJMAIN-3D" x="18161" y="0"></use><use xlink:href="#E369-MJMAIN-31" x="19217" y="0"></use></g></svg></span><script type="math/tex">\delta^{(l)} = (\Theta^{(l)})^T\delta^{(l+1)} .*\ a^{(l)} .*\ (1-a^{(l)}), \ \ a^{(l)}_0 = 1</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="3.052ex" height="2.461ex" viewBox="0 -956.9 1314 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E380-MJMATHI-3B4" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 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y="0"></use><use xlink:href="#E391-MJMAIN-28" x="6652" y="0"></use><use xlink:href="#E391-MJMATHI-74" x="7041" y="0"></use><use xlink:href="#E391-MJMAIN-29" x="7402" y="0"></use></g></svg></span><script type="math/tex">\delta_j^{(l)} = \dfrac{\partial}{\partial z_j^{(l)}} cost(t)</script></p><p>视频内容实际在上文都涉及到了，上节也做了解释：</p><blockquote><p>反向传播算法，即从输出层开始不断<strong>向前迭代</strong>，根据<strong>上一层</strong>的误差依次计算当前层的误差，以求得代价函数的偏导。</p></blockquote><p>不过，这块还是有些不好理解，可回顾视频。</p><p>前文提到输入层没有偏差，所以没有 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.384ex" height="2.461ex" viewBox="0 -956.9 1456.8 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E392-MJMATHI-3B4" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 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y="0"></use></g></g><use xlink:href="#E393-MJMAIN-29" x="20478" y="0"></use><use xlink:href="#E393-MJMAIN-29" x="20867" y="0"></use><use xlink:href="#E393-MJMAIN-29" x="21256" y="0"></use><use xlink:href="#E393-MJSZ2-5D" x="21645" y="-1"></use></g></g></g></g></g></svg></span><script type="math/tex">\begin{gather*} J(\Theta) = - \frac{1}{m} \sum_{i=1}^m \left[y^{(i)} \log ((h_\Theta (x^{(i)}))) + (1 - y^{(i)})\log (1 - (h_\Theta(x^{(i)})))\right] \end{gather*}</script></p><p>再次的，为了方便起见，这里假设样本只有一个，则有：</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="53.269ex" height="2.811ex" viewBox="0 -856.4 22935.2 1210.2" role="img" focusable="false" style="vertical-align: -0.822ex;"><defs><path stroke-width="0" id="E394-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 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style="vertical-align: -1.405ex;"><defs><path stroke-width="0" id="E211-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="E211-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" 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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><path stroke-width="0" id="E396-MJSZ2-28" d="M180 96T180 250T205 541T266 770T353 944T444 1069T527 1150H555Q561 1144 561 1141Q561 1137 545 1120T504 1072T447 995T386 878T330 721T288 513T272 251Q272 133 280 56Q293 -87 326 -209T399 -405T475 -531T536 -609T561 -640Q561 -643 555 -649H527Q483 -612 443 -568T353 -443T266 -270T205 -41Z"></path><path stroke-width="0" id="E396-MJSZ2-29" d="M35 1138Q35 1150 51 1150H56H69Q113 1113 153 1069T243 944T330 771T391 541T416 250T391 -40T330 -270T243 -443T152 -568T69 -649H56Q43 -649 39 -647T35 -637Q65 -607 110 -548Q283 -316 316 56Q324 133 324 251Q324 368 316 445Q278 877 48 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referrerPolicy='no-referrer' /></p><p>再次为了便于计算，我们用到如上图这个三层(输入层一般不计数)神经网络。</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="17.418ex" height="2.461ex" viewBox="0 -956.9 7499.4 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E447-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 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617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E447-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E447-MJMATHI-7A" x="0" y="0"></use><g transform="translate(468,362)"><use transform="scale(0.707)" xlink:href="#E447-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E447-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E447-MJMAIN-29" x="687" y="0"></use></g><use xlink:href="#E447-MJMAIN-3D" x="1607" y="0"></use><g transform="translate(2663,0)"><use xlink:href="#E447-MJMAIN-398" x="0" y="0"></use><g transform="translate(778,362)"><use transform="scale(0.707)" xlink:href="#E447-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E447-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E447-MJMAIN-2212" x="687" y="0"></use><use transform="scale(0.707)" xlink:href="#E447-MJMAIN-31" x="1465" y="0"></use><use transform="scale(0.707)" xlink:href="#E447-MJMAIN-29" x="1964" y="0"></use></g></g><g transform="translate(5205,0)"><use xlink:href="#E447-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,362)"><use transform="scale(0.707)" xlink:href="#E447-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E447-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E447-MJMAIN-2212" x="687" y="0"></use><use transform="scale(0.707)" xlink:href="#E447-MJMAIN-31" x="1465" y="0"></use><use transform="scale(0.707)" xlink:href="#E447-MJMAIN-29" x="1964" y="0"></use></g></g></g></svg></span><script type="math/tex">z^{(l)} = \Theta^{(l-1)}a^{(l-1)}</script>，我们有 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="35.797ex" height="2.928ex" viewBox="0 -956.9 15412.5 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E398-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="E398-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="0" id="E398-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 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y="0"></use><use xlink:href="#E398-MJMATHI-67" x="6775" y="0"></use><use xlink:href="#E398-MJMAIN-28" x="7255" y="0"></use><g transform="translate(7644,0)"><use xlink:href="#E398-MJMATHI-7A" x="0" y="0"></use><g transform="translate(468,362)"><use transform="scale(0.707)" xlink:href="#E398-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E398-MJMAIN-34" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E398-MJMAIN-29" x="888" y="0"></use></g></g><use xlink:href="#E398-MJMAIN-29" x="9117" y="0"></use><use xlink:href="#E398-MJMAIN-3D" x="9784" y="0"></use><use xlink:href="#E398-MJMATHI-67" x="10840" y="0"></use><use xlink:href="#E398-MJMAIN-28" x="11320" y="0"></use><g transform="translate(11709,0)"><use xlink:href="#E398-MJMAIN-398" x="0" y="0"></use><g transform="translate(778,362)"><use transform="scale(0.707)" xlink:href="#E398-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E398-MJMAIN-33" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E398-MJMAIN-29" x="888" y="0"></use></g></g><g transform="translate(13490,0)"><use xlink:href="#E398-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,362)"><use transform="scale(0.707)" xlink:href="#E398-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E398-MJMAIN-33" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E398-MJMAIN-29" x="888" y="0"></use></g></g><use xlink:href="#E398-MJMAIN-29" x="15023" y="0"></use></g></svg></span><script type="math/tex">h_\Theta(x)=a^{(4)}= g(z^{(4)})=g(\Theta^{(3)}a^{(3)})</script></p><p>观察考虑各变量与 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.138ex" height="2.461ex" viewBox="0 -956.9 1781.7 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E411-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="0" id="E411-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="E411-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="0" id="E411-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="#E411-MJMAIN-398" x="0" y="0"></use><g transform="translate(778,362)"><use transform="scale(0.707)" xlink:href="#E411-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E411-MJMAIN-33" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E411-MJMAIN-29" x="888" y="0"></use></g></g></svg></span><script type="math/tex">\Theta^{(3)}</script> 之间的关系，有 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="27.041ex" height="2.928ex" viewBox="0 -956.9 11642.7 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E400-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="E400-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 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xlink:href="#E400-MJMAIN-33" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E400-MJMAIN-29" x="888" y="0"></use></g></g></g></svg></span><script type="math/tex">J(\Theta) \rightarrow  a^{(4)}\rightarrow z^{(4)}\rightarrow \Theta^{(3)}</script></p><p>要计算 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.084ex" height="2.577ex" viewBox="0 -806.1 2189 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E454-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 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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="E411-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="#E411-MJMAIN-398" x="0" y="0"></use><g transform="translate(778,362)"><use transform="scale(0.707)" xlink:href="#E411-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E411-MJMAIN-33" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E411-MJMAIN-29" x="888" y="0"></use></g></g></svg></span><script type="math/tex">\Theta^{(3)}</script> 的微小改变会引起 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transform="scale(0.707)" xlink:href="#E406-MJMAIN-34" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E406-MJMAIN-29" x="888" y="0"></use></g></g></svg></span><script type="math/tex">a^{(4)}</script> 的微小改变又会引起 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.084ex" height="2.577ex" viewBox="0 -806.1 2189 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E407-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="E407-MJMAIN-28" 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749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E407-MJMATHI-4A" x="0" y="0"></use><use xlink:href="#E407-MJMAIN-28" x="633" y="0"></use><use xlink:href="#E407-MJMAIN-398" x="1022" y="0"></use><use xlink:href="#E407-MJMAIN-29" x="1800" y="0"></use></g></svg></span><script type="math/tex"> J(\Theta)</script> 的改变，关系方向也可以反过来写：<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="27.041ex" height="2.928ex" viewBox="0 -956.9 11642.7 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E408-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="0" id="E408-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="E408-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 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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="E408-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="E408-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E408-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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E408-MJMAIN-398" x="0" y="0"></use><g transform="translate(778,362)"><use transform="scale(0.707)" xlink:href="#E408-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E408-MJMAIN-33" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E408-MJMAIN-29" x="888" y="0"></use></g><use xlink:href="#E408-MJMAIN-2192" x="2059" y="0"></use><g transform="translate(3337,0)"><use xlink:href="#E408-MJMATHI-7A" x="0" y="0"></use><g transform="translate(468,362)"><use transform="scale(0.707)" xlink:href="#E408-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E408-MJMAIN-34" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E408-MJMAIN-29" x="888" y="0"></use></g></g><use xlink:href="#E408-MJMAIN-2192" x="5087" y="0"></use><g transform="translate(6365,0)"><use xlink:href="#E408-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,362)"><use transform="scale(0.707)" xlink:href="#E408-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E408-MJMAIN-34" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E408-MJMAIN-29" x="888" y="0"></use></g></g><use xlink:href="#E408-MJMAIN-2192" x="8175" y="0"></use><use xlink:href="#E408-MJMATHI-4A" x="9453" y="0"></use><use xlink:href="#E408-MJMAIN-28" x="10086" y="0"></use><use xlink:href="#E408-MJMAIN-398" x="10475" y="0"></use><use xlink:href="#E408-MJMAIN-29" x="11253" y="0"></use></g></svg></span><script type="math/tex">\Theta^{(3)} \rightarrow z^{(4)} \rightarrow a^{(4)} \rightarrow J(\Theta) </script>。</p><p>如果你还记得微积分（不然你应该也不会看到这里(*<sup>_</sup>\*)~），听起来像不像在暗示链式求导？</p><p>令 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.185ex" height="3.628ex" viewBox="0 -1007.2 6537.8 1562" role="img" focusable="false" style="vertical-align: -1.289ex;"><defs><path stroke-width="0" id="E409-MJMATHI-3B4" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z"></path><path stroke-width="0" id="E409-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="E409-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><path stroke-width="0" id="E409-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="E409-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="E409-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="E409-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="E409-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="E409-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E409-MJMATHI-3B4" x="0" y="0"></use><g transform="translate(453,362)"><use transform="scale(0.707)" xlink:href="#E409-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E409-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E409-MJMAIN-29" x="687" y="0"></use></g><use xlink:href="#E409-MJMAIN-3D" x="1591" y="0"></use><g transform="translate(2369,0)"><g transform="translate(397,0)"><rect stroke="none" width="1461" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E409-MJMAIN-2202" x="749" y="593"></use><g transform="translate(60,-484)"><use transform="scale(0.707)" xlink:href="#E409-MJMAIN-2202" x="0" y="0"></use><g transform="translate(400,0)"><use transform="scale(0.707)" xlink:href="#E409-MJMATHI-7A" x="0" y="0"></use><g transform="translate(331,204)"><use transform="scale(0.5)" xlink:href="#E409-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E409-MJMATHI-6C" x="389" y="0"></use><use transform="scale(0.5)" xlink:href="#E409-MJMAIN-29" x="687" y="0"></use></g></g></g></g></g><use xlink:href="#E409-MJMATHI-4A" x="4348" y="0"></use><use xlink:href="#E409-MJMAIN-28" x="4981" y="0"></use><use xlink:href="#E409-MJMAIN-398" x="5370" y="0"></use><use xlink:href="#E409-MJMAIN-29" x="6148" y="0"></use></g></svg></span><script type="math/tex">\delta^{(l)} = \frac{\partial}{\partial z^{(l)}} J(\Theta)</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="5.084ex" height="2.577ex" viewBox="0 -806.1 2189 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E454-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="E454-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="E454-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="0" id="E454-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="#E454-MJMATHI-4A" x="0" y="0"></use><use xlink:href="#E454-MJMAIN-28" x="633" y="0"></use><use xlink:href="#E454-MJMAIN-398" x="1022" y="0"></use><use xlink:href="#E454-MJMAIN-29" x="1800" y="0"></use></g></svg></span><script type="math/tex">J(\Theta)</script> 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="4.138ex" height="2.461ex" viewBox="0 -956.9 1781.7 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E411-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="0" id="E411-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="E411-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="0" id="E411-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="#E411-MJMAIN-398" x="0" y="0"></use><g transform="translate(778,362)"><use transform="scale(0.707)" xlink:href="#E411-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E411-MJMAIN-33" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E411-MJMAIN-29" x="888" y="0"></use></g></g></svg></span><script type="math/tex">\Theta^{(3)}</script> 的偏导：</p><p><span 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y="0"></use></g></g></g></g></g><use xlink:href="#E412-MJMAIN-3D" x="10150" y="0"></use><g transform="translate(11206,0)"><use xlink:href="#E412-MJMATHI-3B4" x="0" y="0"></use><g transform="translate(453,362)"><use transform="scale(0.707)" xlink:href="#E412-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E412-MJMAIN-34" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E412-MJMAIN-29" x="888" y="0"></use></g></g><g transform="translate(12663,0)"><g transform="translate(120,0)"><rect stroke="none" width="1780" height="60" x="0" y="220"></rect><g transform="translate(169,419)"><use transform="scale(0.707)" xlink:href="#E412-MJMAIN-2202" x="0" y="0"></use><g transform="translate(400,0)"><use transform="scale(0.707)" xlink:href="#E412-MJMATHI-7A" x="0" y="0"></use><g transform="translate(331,256)"><use transform="scale(0.5)" xlink:href="#E412-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E412-MJMAIN-34" x="389" y="0"></use><use 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y="0"></use><g transform="translate(329,204)"><use transform="scale(0.5)" xlink:href="#E418-MJMATHI-7A" x="0" y="0"></use><g transform="translate(234,178)"><use transform="scale(0.5)" xlink:href="#E418-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E418-MJMAIN-34" x="389" y="0"></use><use transform="scale(0.5)" xlink:href="#E418-MJMAIN-29" x="889" y="0"></use></g></g></g></g></g></g><use xlink:href="#E418-MJMAIN-3D" x="18840" y="0"></use><use xlink:href="#E418-MJMATHI-67" x="19896" y="0"></use><use xlink:href="#E418-MJMAIN-28" x="20376" y="0"></use><g transform="translate(20765,0)"><use xlink:href="#E418-MJMATHI-7A" x="0" y="0"></use><g transform="translate(468,362)"><use transform="scale(0.707)" xlink:href="#E418-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E418-MJMAIN-34" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E418-MJMAIN-29" x="888" y="0"></use></g></g><use xlink:href="#E418-MJMAIN-29" x="22237" y="0"></use><use xlink:href="#E418-MJMAIN-2212" x="22848" y="0"></use><use xlink:href="#E418-MJMATHI-79" x="23849" y="0"></use><use xlink:href="#E418-MJMAIN-3D" x="24623" y="0"></use><g transform="translate(25679,0)"><use xlink:href="#E418-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,362)"><use transform="scale(0.707)" xlink:href="#E418-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E418-MJMAIN-34" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E418-MJMAIN-29" x="888" y="0"></use></g></g><use xlink:href="#E418-MJMAIN-2212" x="27434" y="0"></use><use xlink:href="#E418-MJMATHI-79" x="28434" y="0"></use></g></svg></span><script type="math/tex">\delta^{(4)}=\frac{\partial}{\partial z^{(4)}}J(\Theta)={{y}}\frac{-e^{-z^{(4)}}}{1+e^{-z^{(4)}}}+\left( 1-{{y}} \right)\frac{{e^{z^{(4)}}}}{1+e^{z^{(4)}}} = g(z^{(4)}) - y = a^{(4)}-y</script></p><p>即证得 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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y="0"></use><use transform="scale(0.707)" xlink:href="#E419-MJMAIN-34" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E419-MJMAIN-29" x="888" y="0"></use></g></g><use xlink:href="#E419-MJMAIN-2212" x="4545" y="0"></use><use xlink:href="#E419-MJMATHI-79" x="5545" y="0"></use></g></svg></span><script type="math/tex">\delta^{(4)} = a^{(4)}-y</script></p><p>对于任意的输出层 <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 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id="MathJax-Element-328">\frac{\partial}{\partial\Theta^{(L-1)}} J(\Theta) =  a^{(L-1)}\delta^{(L)}, \ \ \delta^{(L)} = a^{(L)}-y</script></div></div><p>好了，接下来来看一下 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.084ex" height="2.577ex" viewBox="0 -806.1 2189 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E454-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="E454-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 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x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E433-MJMAIN-29" x="888" y="0"></use></g></g><use xlink:href="#E433-MJMAIN-2E" x="4710" y="0"></use><use xlink:href="#E433-MJMAIN-2217" x="5155" y="0"></use><use xlink:href="#E433-MJMAIN-28" x="5905" y="0"></use><use xlink:href="#E433-MJMAIN-31" x="6294" y="0"></use><use xlink:href="#E433-MJMAIN-2212" x="7016" y="0"></use><g transform="translate(8017,0)"><use xlink:href="#E433-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,362)"><use transform="scale(0.707)" xlink:href="#E433-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E433-MJMAIN-33" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E433-MJMAIN-29" x="888" y="0"></use></g></g><use xlink:href="#E433-MJMAIN-29" x="9549" y="0"></use></g></svg></span><script type="math/tex">\frac{\partial a^{(3)}}{\partial z^{(3)}}=a^{(3)} .*\ (1-a^{(3)})</script>，</p><blockquote><p>证明时为先求导后添加偏置单元，与前向传播算法顺序一致，实际实现时，求导和添加偏置单元的顺序可作调换，由于一般选择忽略偏置单元的误差，所以并不影响结果。</p></blockquote><p>即证得 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="57.734ex" height="2.928ex" viewBox="0 -956.9 24857.5 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E434-MJMATHI-3B4" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z"></path><path stroke-width="0" id="E434-MJMAIN-28" d="M94 250Q94 319 104 381T127 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x="1022" y="0"></use><use xlink:href="#E454-MJMAIN-29" x="1800" y="0"></use></g></svg></span><script type="math/tex">J(\Theta)</script> 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="4.138ex" height="2.461ex" viewBox="0 -956.9 1781.7 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E441-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="0" id="E441-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="E441-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="E441-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="#E441-MJMAIN-398" x="0" y="0"></use><g transform="translate(778,362)"><use transform="scale(0.707)" xlink:href="#E441-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E441-MJMAIN-31" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E441-MJMAIN-29" x="888" y="0"></use></g></g></svg></span><script type="math/tex">\Theta^{(1)}</script> 的偏导，是不是能得到同样的结果？</p><h2><a name='header-n138' class='md-header-anchor '></a>9.4 实现注意点: 参数展开(Implementation Note: Unrolling Parameters)</h2><p>在 Octave/Matlab 中，如果要使用类似于 <code>fminunc</code> 等高级最优化函数，其函数参数、函数返回值等都为且只为向量，而由于神经网络中的权重是多维矩阵，所以需要用到参数展开这个技巧。</p><p>说白了，这个技巧就是把多个矩阵转换为一个长长的向量，便于传入函数，之后再根据矩阵维度，转回矩阵即可。</p><p>Octave/Matlab 代码：</p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="octave"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="octave"><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>9</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;"><span class="cm-comment">% 多个矩阵展开为一个向量</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">Theta1</span> = <span class="cm-builtin">ones</span>(<span class="cm-number">11</span>, <span class="cm-number">10</span>); &nbsp; &nbsp;<span class="cm-comment">% 创建维度为 11 * 10 的矩阵</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;"><span class="cm-variable">Theta2</span> = <span class="cm-builtin">ones</span>(<span class="cm-number">2</span>, <span class="cm-number">4</span>) <span class="cm-operator">*</span> <span class="cm-number">2</span>; &nbsp;<span class="cm-comment">% 创建维度为 2 * 4 的矩阵</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;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">ThetaVec</span> = [<span class="cm-variable">Theta1</span>(:); <span class="cm-variable">Theta2</span>(:)]; <span class="cm-comment">% 将上面两个矩阵展开为向量</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;">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;"><span class="cm-comment">% 从一个向量重构还原回多个矩阵</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;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">Theta1</span> = <span class="cm-builtin">reshape</span>(<span class="cm-variable">ThetaVec</span>(<span class="cm-number">1</span>:<span class="cm-number">110</span>), <span class="cm-number">11</span>, <span class="cm-number">10</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;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">Theta2</span> = <span class="cm-builtin">reshape</span>(<span class="cm-variable">ThetaVec</span>(<span class="cm-number">111</span>:<span class="cm-number">118</span>), <span class="cm-number">2</span>, <span class="cm-number">4</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;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Theta2 = reshape(ThetaVec(111:(111 + 2 * 4) - 1), 2, 4)</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 207px;"></div><div class="CodeMirror-gutters" style="height: 207px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 27px;"></div></div></div></div></pre><blockquote><p><code>reshape(A,m,n)</code>: 将向量 A 重构为 m * n 维矩阵。</p></blockquote><h2><a name='header-n145' class='md-header-anchor '></a>9.5 梯度检验(Gradient Checking)</h2><p>由于神经网络模型中的反向传播算法较为复杂，在小细节非常容易出错，从而无法得到最优解，故引入梯度检验。</p><p>梯度检验采用数值估算(Numerical estimation)梯度的方法，被用于验证反向传播算法的正确性。</p><p><img src='images/20180125_162704.png' alt='' referrerPolicy='no-referrer' /></p><p>把视 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.807ex" height="2.11ex" viewBox="0 -806.1 778 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E279-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" 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transform="translate(4035,-686)"><use xlink:href="#E442-MJMAIN-32" x="0" y="0"></use><use xlink:href="#E442-MJMATHI-3F5" x="500" y="0"></use></g></g></g></g></svg></span><script type="math/tex">\dfrac{\partial}{\partial\Theta}J(\Theta) \approx \dfrac{J(\Theta + \epsilon) - J(\Theta - \epsilon)}{2\epsilon}</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="0.943ex" height="1.41ex" viewBox="0 -504.6 406 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E452-MJMATHI-3F5" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E452-MJMATHI-3F5" x="0" y="0"></use></g></svg></span><script type="math/tex">\epsilon</script> 为极小值，由于太小时容易出现数值运算问题，一般取 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.654ex" height="2.461ex" viewBox="0 -956.9 2003.7 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E444-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="E444-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 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y="0"></use></g></g></svg></span><script type="math/tex">10^{-4}</script>。</p><p>&nbsp;</p><p>对于矩阵 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.807ex" height="2.11ex" viewBox="0 -806.1 778 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E279-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E279-MJMAIN-398" x="0" y="0"></use></g></svg></span><script 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x="16457" y="0"></use><g transform="translate(16902,0)"><use xlink:href="#E445-MJMAIN-398" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E445-MJMATHI-6A" x="1100" y="-213"></use></g><use xlink:href="#E445-MJMAIN-2212" x="18294" y="0"></use><use xlink:href="#E445-MJMATHI-3F5" x="19294" y="0"></use><use xlink:href="#E445-MJMAIN-2C" x="19700" y="0"></use><use xlink:href="#E445-MJMAIN-2026" x="20145" y="0"></use><use xlink:href="#E445-MJMAIN-2C" x="21483" y="0"></use><g transform="translate(21928,0)"><use xlink:href="#E445-MJMAIN-398" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E445-MJMATHI-6E" x="1100" y="-213"></use></g><use xlink:href="#E445-MJMAIN-29" x="23230" y="0"></use></g><g transform="translate(11416,-686)"><use xlink:href="#E445-MJMAIN-32" x="0" y="0"></use><use xlink:href="#E445-MJMATHI-3F5" x="500" y="0"></use></g></g></g></g></svg></span><script type="math/tex">\dfrac{\partial}{\partial\Theta_j}J(\Theta) \approx \dfrac{J(\Theta_1, \dots, \Theta_j + \epsilon, \dots, \Theta_n) - J(\Theta_1, \dots, \Theta_j - \epsilon, \dots, \Theta_n)}{2\epsilon}</script></p><p>Octave/Matlab 代码：</p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="octave"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="octave"><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;"><span class="cm-variable">epsilon</span> = <span class="cm-number">1e-4</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-keyword">for</span> <span class="cm-variable">i</span> = <span class="cm-number">1</span>:<span class="cm-variable">n</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;<span class="cm-variable">thetaPlus</span> = <span class="cm-variable">theta</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;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-variable">thetaPlus</span>(<span class="cm-variable">i</span>) <span class="cm-operator">+</span>= <span class="cm-variable">epsilon</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;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-variable">thetaMinus</span> = <span class="cm-variable">theta</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;"> &nbsp;<span class="cm-variable">thetaMinus</span>(<span class="cm-variable">i</span>) <span class="cm-operator">-</span>= <span class="cm-variable">epsilon</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;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;<span class="cm-variable">gradApprox</span>(<span class="cm-variable">i</span>) = (<span class="cm-variable">J</span>(<span class="cm-variable">thetaPlus</span>) <span class="cm-operator">-</span> <span class="cm-variable">J</span>(<span class="cm-variable">thetaMinus</span>))<span class="cm-operator">/</span>(<span class="cm-number">2</span><span class="cm-operator">*</span><span class="cm-variable">epsilon</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;"><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: 184px;"></div><div class="CodeMirror-gutters" style="height: 184px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 27px;"></div></div></div></div></pre><p>在得出 gradApprox 梯度向量后，将其同之前计算的偏导 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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比较，如果相等或很接近，即说明算法没有问题。</p><p>在确认算法<strong>没有问题后</strong>（一般只需运行一次），由于数值估计的梯度检验效率很低，所以一定要<strong>禁用它</strong>。</p><h2><a name='header-n157' class='md-header-anchor '></a>9.6 随机初始化(Random Initialization)</h2><p>逻辑回归中，初始参数向量全为 0 没什么问题，在神经网络中，情况就不一样了。</p><p>初始权重如果全为 0，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="17.418ex" height="2.461ex" viewBox="0 -956.9 7499.4 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E447-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 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transform="scale(0.707)" xlink:href="#E450-MJMAIN-29" x="687" y="0"></use></g><g transform="translate(778,-303)"><use transform="scale(0.707)" xlink:href="#E450-MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E450-MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#E450-MJMATHI-6A" x="623" y="0"></use></g><use xlink:href="#E450-MJMAIN-2208" x="1916" y="0"></use><use xlink:href="#E450-MJMAIN-5B" x="2861" y="0"></use><use xlink:href="#E450-MJMAIN-2212" x="3139" y="0"></use><use xlink:href="#E450-MJMATHI-3F5" x="3917" y="0"></use><use xlink:href="#E450-MJMAIN-2C" x="4323" y="0"></use><use xlink:href="#E450-MJMATHI-3F5" x="4768" y="0"></use><use xlink:href="#E450-MJMAIN-5D" x="5174" y="0"></use></g></svg></span><script type="math/tex">\Theta^{(l)}_{i,j} \in [-\epsilon, \epsilon]</script>。</p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="octave"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="octave"><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>5</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;"><span class="cm-variable">If</span> <span class="cm-variable">the</span> <span class="cm-variable">dimensions</span> <span class="cm-variable">of</span> <span class="cm-variable">Theta1</span> <span class="cm-variable">is</span> <span class="cm-number">10</span><span class="cm-variable">x11</span>, <span class="cm-variable">Theta2</span> <span class="cm-variable">is</span> <span class="cm-number">10</span><span class="cm-variable">x11</span> <span class="cm-variable">and</span> <span class="cm-variable">Theta3</span> <span class="cm-variable">is</span> <span class="cm-number">1</span><span class="cm-variable">x11</span><span class="cm-error">.</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 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;"><span class="cm-variable">Theta1</span> = <span class="cm-builtin">rand</span>(<span class="cm-number">10</span>,<span class="cm-number">11</span>) <span class="cm-operator">*</span> (<span class="cm-number">2</span> <span class="cm-operator">*</span> <span class="cm-variable">INIT_EPSILON</span>) <span class="cm-operator">-</span> <span class="cm-variable">INIT_EPSILON</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;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">Theta2</span> = <span class="cm-builtin">rand</span>(<span class="cm-number">10</span>,<span class="cm-number">11</span>) <span class="cm-operator">*</span> (<span class="cm-number">2</span> <span class="cm-operator">*</span> <span class="cm-variable">INIT_EPSILON</span>) <span class="cm-operator">-</span> <span class="cm-variable">INIT_EPSILON</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;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">Theta3</span> = <span class="cm-builtin">rand</span>(<span class="cm-number">1</span>,<span class="cm-number">11</span>) <span class="cm-operator">*</span> (<span class="cm-number">2</span> <span class="cm-operator">*</span> <span class="cm-variable">INIT_EPSILON</span>) <span class="cm-operator">-</span> <span class="cm-variable">INIT_EPSILON</span>;</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 115px;"></div><div class="CodeMirror-gutters" style="height: 115px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 27px;"></div></div></div></div></pre><blockquote><p><code>rand(m,n)</code>: 返回一个在区间 (0,1) 内均匀分布的随机矩阵。</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="0.943ex" height="1.41ex" viewBox="0 -504.6 406 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E452-MJMATHI-3F5" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E452-MJMATHI-3F5" x="0" y="0"></use></g></svg></span><script type="math/tex">\epsilon</script>: 和梯度下降中的 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.943ex" height="1.41ex" viewBox="0 -504.6 406 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E452-MJMATHI-3F5" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E452-MJMATHI-3F5" x="0" y="0"></use></g></svg></span><script type="math/tex">\epsilon</script> 没有联系，这里只是一个任意实数，给定了权重矩阵初始化值的范围。</p></blockquote><h2><a name='header-n167' class='md-header-anchor '></a>9.7 综合起来(Putting It Together)</h2><p>一般来说，应用神经网络有如下步骤：</p><ol start='' ><li><p>神经网络的建模（后续补充）</p><ul><li>选取特征，确定特征向量 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.329ex" height="1.41ex" viewBox="0 -504.6 572 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E12-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="#E12-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex">x</script> 的维度，即输入单元的数量。</li><li>鉴别分类，确定预测向量 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.983ex" height="2.577ex" viewBox="0 -806.1 2576.1 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E326-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="E326-MJMAIN-398" d="M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z"></path><path stroke-width="0" id="E326-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 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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="#E326-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E326-MJMAIN-398" x="814" y="-218"></use><use xlink:href="#E326-MJMAIN-28" x="1226" y="0"></use><use xlink:href="#E326-MJMATHI-78" x="1615" y="0"></use><use xlink:href="#E326-MJMAIN-29" x="2187" y="0"></use></g></svg></span><script type="math/tex">h_\Theta(x)</script> 的维度，即输出单元的数量。</li><li>确定隐藏层有几层以及每层隐藏层有多少个隐藏单元。</li></ul><blockquote><p>默认情况下，隐藏层至少要有一层，也可以有多层，层数越多一般意味着效果越好，计算量越大。</p></blockquote></li><li><p>训练神经网络</p><ol start='' ><li><p>随机初始化初始权重矩阵</p></li><li><p>应用前向传播算法计算初始预测</p></li><li><p>计算代价函数 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.084ex" height="2.577ex" viewBox="0 -806.1 2189 1109.7" role="img" focusable="false" style="vertical-align: 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y="0"></use></g></svg></span><script type="math/tex">J(\Theta)</script> 的偏导数</p></li><li><p>使用梯度检验检查算法的正确性，别忘了用完就禁用它</p></li><li><p>丢给最优化函数最小化代价函数</p><blockquote><p>由于神经网络的代价函数非凸，最优化时不一定会收敛在全局最小值处，高级最优化函数能确保收敛在某个<strong>局部</strong>最小值处。</p></blockquote></li></ol></li></ol><p>&nbsp;</p><h2><a name='header-n199' class='md-header-anchor '></a>9.8 自主驾驶(Autonomous Driving)</h2><p><img src='images/20180125_195029.png' alt='' referrerPolicy='no-referrer' /></p><p>描述了神经网络在于<a href='https://www.coursera.org/learn/machine-learning/lecture/zYS8T/autonomous-driving'>自动驾驶</a>领域的应用实例，用于打鸡血，笔记略。</p></div>

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