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<!doctype html>
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</style><title>README</title>
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<body class='typora-export os-windows'><div class='typora-export-content'>
<div id='write' class=''><h1 id='back-propagation-neural-network'><span>Back-Propagation-Neural-Network</span></h1><p><a href='https://github.com/GavinTechStudio/Back-Propagation-Neural-Network'><img src="https://badgen.net/badge/github/BPNN?icon&label=GitHub" referrerpolicy="no-referrer" alt="BPNN"></a><span> </span><a href='https://github.com/topics/cpp'><img src="https://img.shields.io/badge/support-C%2B%2B11%20or%20later-blue?style=flat&logo=cplusplus" referrerpolicy="no-referrer" alt="C++"></a><span> </span><a href='https://cmake.org/'><img src="https://img.shields.io/badge/support-v2.8.12%20or%20later-blue?style=flat&logo=cmake" referrerpolicy="no-referrer" alt="CMake"></a><span> </span><a href='https://github.com/GavinTechStudio/Back-Propagation-Neural-Network/commits'><img src="https://img.shields.io/github/last-commit/GavinTechStudio/Back-Propagation-Neural-Network" referrerpolicy="no-referrer" alt="update"></a><span> </span><a href='https://github.com/GavinTechStudio/Back-Propagation-Neural-Network/actions/workflows/pages/pages-build-deployment'><img src="https://github.com/GavinTechStudio/Back-Propagation-Neural-Network/actions/workflows/pages/pages-build-deployment/badge.svg" referrerpolicy="no-referrer" alt="pages-build-deployment"></a><span> </span></p><p><a href='https://github.com/GavinTechStudio/Back-Propagation-Neural-Network'><span>本项目</span></a><span>是对项目 </span><a href='https://github.com/GavinTechStudio/bpnn_with_cpp'><strong><span>GavinTechStudio/bpnn_with_cpp</span></strong></a><span> 的代码重构，基于C++实现基础BP神经网络，有助于深入理解BP神经网络原理。</span></p><h2 id='项目结构'><span>项目结构</span></h2><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang=""><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.5px; left: 8px;"><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: 0px; 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><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: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">.</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">├── CMakeLists.txt</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">├── data</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">│ ├── testdata.txt</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">│ └── traindata.txt</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">├── lib</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">│ ├── Config.h</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">│ ├── Net.cpp</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">│ ├── Net.h</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">│ ├── Utils.cpp</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">│ └── Utils.h</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">└── main.cpp</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 276px;"></div><div class="CodeMirror-gutters" style="display: none; height: 276px;"></div></div></div></pre><h4 id='主要框架'><span>主要框架</span></h4><ul><li><p><span>Net：网络具体实现</span></p></li><li><p><span>Config：网络参数设置</span></p></li><li><p><span>Utils：工具类</span></p><ul><li><span>数据加载</span></li><li><span>激活函数</span></li></ul></li><li><p><span>main：网络具体应用</span></p></li></ul><h2 id='训练原理'><span>训练原理</span></h2><blockquote><p><span>具体公式推导请看视频讲解</span><a href='https://www.bilibili.com/video/BV1Y64y1z7jM?p=1'><span>彻底搞懂BP神经网络 理论推导+代码实现（C++）</span></a></p></blockquote><h4 id='注意本部分文档包含大量数学公式由于github-markdown不支持数学公式渲染推荐以下阅读方式'><span>注意：本部分文档包含大量数学公式，由于GitHub markdown不支持数学公式渲染，推荐以下阅读方式：</span></h4><ol start='' ><li><span>如果您使用的是Chrome、Edge、Firefox等浏览器，可以安装插件</span><a href='https://chrome.google.com/webstore/detail/mathjax-plugin-for-github/ioemnmodlmafdkllaclgeombjnmnbima'><span>MathJax Plugin for Github</span></a><span>（需要网络能够访问chrome web store）。</span></li><li><span>使用</span><a href='https://gavintechstudio.github.io/Back-Propagation-Neural-Network/README.pdf'><strong><span>PDF</span></strong></a><span>的方式进行阅读（</span><strong><span>推荐</span></strong><span>）。</span></li><li><span>使用</span><a href='https://gavintechstudio.github.io/Back-Propagation-Neural-Network/README.html'><strong><span>预渲染的静态网页</span></strong></a><span>进行阅读（</span><strong><span>推荐</span></strong><span>）。</span></li><li><span>按</span><code>.</code><span>键或</span><a href='https://github.dev/GavinTechStudio/Back-Propagation-Neural-Network'><span>点击链接</span></a><span>进入GitHub在线IDE预览</span><code>README.md</code><span>文件。</span></li></ol><h3 id='0-神经网络结构图'><span>0. 神经网络结构图</span></h3><p><img src="img/net-info.png" referrerpolicy="no-referrer"></p><h3 id='1-forward前向传播）'><span>1. Forward（前向传播）</span></h3><h4 id='11-输入层向隐藏层传播'><span>1.1 输入层向隐藏层传播</span></h4><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n455" cid="n455" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="21.809ex" height="4.847ex" role="img" focusable="false" viewBox="0 -950 9639.6 2142.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.697ex;"><defs><path id="MJX-121-TEX-I-210E" 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 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xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-88-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-88-TEX-I-1D456"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">i</script><span>个输入层节点的值，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.019ex" height="1.668ex" role="img" focusable="false" viewBox="0 -443 1334.3 737.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.666ex;"><defs><path id="MJX-57-TEX-I-1D464" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z"></path><path id="MJX-57-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-57-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D464" xlink:href="#MJX-57-TEX-I-1D464"></use></g><g data-mml-node="TeXAtom" transform="translate(749,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-57-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(345,0)"><use data-c="1D457" xlink:href="#MJX-57-TEX-I-1D457"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">w_{ij}</script><span>为第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.781ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 345 672" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-88-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-88-TEX-I-1D456"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">i</script><span>个输入层节点到第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.932ex" height="1.957ex" role="img" focusable="false" viewBox="0 -661 412 865" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.462ex;"><defs><path id="MJX-84-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D457" xlink:href="#MJX-84-TEX-I-1D457"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">j</script><span>个隐藏层节点的权重，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.127ex" height="2.261ex" role="img" focusable="false" viewBox="0 -705 940.3 999.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.666ex;"><defs><path id="MJX-60-TEX-I-1D6FD" d="M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z"></path><path id="MJX-60-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D6FD" xlink:href="#MJX-60-TEX-I-1D6FD"></use></g><g data-mml-node="mi" transform="translate(599,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-60-TEX-I-1D457"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>j</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\beta_j</script><span>为第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.932ex" height="1.957ex" role="img" focusable="false" viewBox="0 -661 412 865" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.462ex;"><defs><path id="MJX-84-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D457" xlink:href="#MJX-84-TEX-I-1D457"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">j</script><span>个隐藏层节点偏置值，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.346ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1921 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-72-TEX-I-1D70E" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></path><path id="MJX-72-TEX-N-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 id="MJX-72-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-72-TEX-N-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="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D70E" xlink:href="#MJX-72-TEX-I-1D70E"></use></g><g data-mml-node="mo" transform="translate(571,0)"><use data-c="28" xlink:href="#MJX-72-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(960,0)"><use data-c="1D465" xlink:href="#MJX-72-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1532,0)"><use data-c="29" xlink:href="#MJX-72-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\sigma(x)</script><span>为</span><strong><span>Sigmoid</span></strong><span>激活函数，后续也将继续用这个表达，其表达式如下</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n457" cid="n457" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="15.657ex" height="4.774ex" role="img" focusable="false" viewBox="0 -1342 6920.6 2110" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -1.738ex;"><defs><path id="MJX-122-TEX-I-1D70E" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></path><path id="MJX-122-TEX-N-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 id="MJX-122-TEX-I-1D465" 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 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transform="translate(1722.4,0)"><g data-mml-node="mi"><use data-c="1D452" xlink:href="#MJX-122-TEX-I-1D452"></use></g><g data-mml-node="TeXAtom" transform="translate(499,289) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-122-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(778,0)"><use data-c="1D465" xlink:href="#MJX-122-TEX-I-1D465"></use></g></g></g></g><rect width="3426" height="60" x="120" y="220"></rect></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>σ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>x</mi></mrow></msup></mrow></mfrac></math></mjx-assistive-mml></mjx-container></div></div><p><span>本项目中的代码实现如下：</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="C++"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="c++"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.5px; left: 8px;"><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: 0px; 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><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: 0px; width: 0px;"></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-3">size_t</span> <span class="cm-variable">j</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">j</span> <span class="cm-operator"><</span> <span class="cm-variable">Config::HIDENODE</span>; <span class="cm-operator">++</span><span class="cm-variable">j</span>) {</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-variable-3">double</span> <span class="cm-variable">sum</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">for</span> (<span class="cm-variable-3">size_t</span> <span class="cm-variable">i</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">i</span> <span class="cm-operator"><</span> <span class="cm-variable">Config::INNODE</span>; <span class="cm-operator">++</span><span class="cm-variable">i</span>) {</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">sum</span> <span class="cm-operator">+=</span> <span class="cm-variable">inputLayer</span>[<span class="cm-variable">i</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span> <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">inputLayer</span>[<span class="cm-variable">i</span>]<span class="cm-operator">-></span><span class="cm-variable">weight</span>[<span class="cm-variable">j</span>];</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> }</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">sum</span> <span class="cm-operator">-=</span> <span class="cm-variable">hiddenLayer</span>[<span class="cm-variable">j</span>]<span class="cm-operator">-></span><span class="cm-variable">bias</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">hiddenLayer</span>[<span class="cm-variable">j</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span> <span class="cm-operator">=</span> <span class="cm-variable">Utils::sigmoid</span>(<span class="cm-variable">sum</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">}</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 230px;"></div><div class="CodeMirror-gutters" style="display: none; height: 230px;"></div></div></div></pre><h4 id='12-隐藏层向输出层传播'><span>1.2 隐藏层向输出层传播</span></h4><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n461" cid="n461" mdtype="math_block" 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transform="translate(412,0)"><use data-c="1D458" xlink:href="#MJX-123-TEX-I-1D458"></use></g></g></g><g data-mml-node="mo" transform="translate(7245.9,0)"><use data-c="2212" xlink:href="#MJX-123-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(8246.1,0)"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-123-TEX-I-1D706"></use></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-123-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(9280.5,0)"><use data-c="29" xlink:href="#MJX-123-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo>=</mo><mi>σ</mi><mo stretchy="false">(</mo><munder><mo data-mjx-texclass="OP">∑</mo><mi>j</mi></munder><msub><mi>h</mi><mi>j</mi></msub><msub><mi>v</mi><mrow data-mjx-texclass="ORD"><mi>j</mi><mi>k</mi></mrow></msub><mo>−</mo><msub><mi>λ</mi><mi>k</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>其中</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.13ex" height="2.296ex" role="img" focusable="false" viewBox="0 -810 941.4 1015" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-75-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 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unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\hat{y_k}</script><span>为第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 521 705" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-85-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-85-TEX-I-1D458"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">k</script><span>个输出层节点的值（预测值），</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.15ex" height="2.236ex" role="img" focusable="false" viewBox="0 -694 950.3 988.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.666ex;"><defs><path id="MJX-80-TEX-I-210E" 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 id="MJX-80-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="210E" xlink:href="#MJX-80-TEX-I-210E"></use></g><g data-mml-node="mi" transform="translate(609,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-80-TEX-I-1D457"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>j</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">h_j</script><span>为第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.932ex" height="1.957ex" role="img" focusable="false" viewBox="0 -661 412 865" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.462ex;"><defs><path id="MJX-84-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D457" xlink:href="#MJX-84-TEX-I-1D457"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">j</script><span>个隐藏层节点的值，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.778ex" height="1.668ex" role="img" focusable="false" viewBox="0 -443 1227.7 737.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.666ex;"><defs><path id="MJX-83-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path id="MJX-83-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path><path id="MJX-83-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-83-TEX-I-1D463"></use></g><g data-mml-node="TeXAtom" transform="translate(518,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D457" xlink:href="#MJX-83-TEX-I-1D457"></use></g><g data-mml-node="mi" transform="translate(412,0)"><use data-c="1D458" xlink:href="#MJX-83-TEX-I-1D458"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mrow data-mjx-texclass="ORD"><mi>j</mi><mi>k</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">v_{jk}</script><span>为第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.932ex" height="1.957ex" role="img" focusable="false" viewBox="0 -661 412 865" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.462ex;"><defs><path id="MJX-84-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D457" xlink:href="#MJX-84-TEX-I-1D457"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">j</script><span>个隐藏层节点到第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 521 705" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-85-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-85-TEX-I-1D458"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">k</script><span>个输出层节点的权重，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.34ex" height="1.927ex" role="img" focusable="false" viewBox="0 -694 1034.4 851.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-70-TEX-I-1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path id="MJX-70-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-70-TEX-I-1D706"></use></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-70-TEX-I-1D458"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mi>k</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\lambda_k</script><span>为第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 521 705" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-85-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-85-TEX-I-1D458"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">k</script><span>个输出层节点的偏置值，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.346ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1921 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-72-TEX-I-1D70E" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></path><path id="MJX-72-TEX-N-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 id="MJX-72-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-72-TEX-N-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="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D70E" xlink:href="#MJX-72-TEX-I-1D70E"></use></g><g data-mml-node="mo" transform="translate(571,0)"><use data-c="28" xlink:href="#MJX-72-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(960,0)"><use data-c="1D465" xlink:href="#MJX-72-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1532,0)"><use data-c="29" xlink:href="#MJX-72-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\sigma(x)</script><span>为激活函数。</span></p><p><span>本项目中的代码实现如下：</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="C++"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="c++"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.5px; left: 8px;"><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: 0px; 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><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: 0px; width: 0px;"></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-3">size_t</span> <span class="cm-variable">k</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">k</span> <span class="cm-operator"><</span> <span class="cm-variable">Config::OUTNODE</span>; <span class="cm-operator">++</span><span class="cm-variable">k</span>) {</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable-3">double</span> <span class="cm-variable">sum</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">for</span> (<span class="cm-variable-3">size_t</span> <span class="cm-variable">j</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">j</span> <span class="cm-operator"><</span> <span class="cm-variable">Config::HIDENODE</span>; <span class="cm-operator">++</span><span class="cm-variable">j</span>) {</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">sum</span> <span class="cm-operator">+=</span> <span class="cm-variable">hiddenLayer</span>[<span class="cm-variable">j</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span> <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">hiddenLayer</span>[<span class="cm-variable">j</span>]<span class="cm-operator">-></span><span class="cm-variable">weight</span>[<span class="cm-variable">k</span>];</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> }</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">sum</span> <span class="cm-operator">-=</span> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">bias</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span> <span class="cm-operator">=</span> <span class="cm-variable">Utils::sigmoid</span>(<span class="cm-variable">sum</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">}</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 230px;"></div><div class="CodeMirror-gutters" style="display: none; height: 230px;"></div></div></div></pre><h3 id='2-计算损失函数loss-function）'><span>2. 计算损失函数（Loss Function）</span></h3><p><span>损失函数定义如下：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n467" cid="n467" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="23.321ex" height="5.786ex" role="img" focusable="false" viewBox="0 -1342 10308 2557.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.75ex;"><defs><path id="MJX-124-TEX-I-1D43F" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path><path id="MJX-124-TEX-I-1D45C" d="M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></path><path id="MJX-124-TEX-I-1D460" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path id="MJX-124-TEX-N-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 id="MJX-124-TEX-N-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 id="MJX-124-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-124-TEX-LO-2211" d="M60 948Q63 950 665 950H1267L1325 815Q1384 677 1388 669H1348L1341 683Q1320 724 1285 761Q1235 809 1174 838T1033 881T882 898T699 902H574H543H251L259 891Q722 258 724 252Q725 250 724 246Q721 243 460 -56L196 -356Q196 -357 407 -357Q459 -357 548 -357T676 -358Q812 -358 896 -353T1063 -332T1204 -283T1307 -196Q1328 -170 1348 -124H1388Q1388 -125 1381 -145T1356 -210T1325 -294L1267 -449L666 -450Q64 -450 61 -448Q55 -446 55 -439Q55 -437 57 -433L590 177Q590 178 557 222T452 366T322 544L56 909L55 924Q55 945 60 948Z"></path><path id="MJX-124-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-124-TEX-N-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 id="MJX-124-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-124-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-124-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path><path id="MJX-124-TEX-N-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="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D43F" xlink:href="#MJX-124-TEX-I-1D43F"></use></g><g data-mml-node="mi" transform="translate(681,0)"><use data-c="1D45C" xlink:href="#MJX-124-TEX-I-1D45C"></use></g><g data-mml-node="mi" transform="translate(1166,0)"><use data-c="1D460" xlink:href="#MJX-124-TEX-I-1D460"></use></g><g data-mml-node="mi" transform="translate(1635,0)"><use data-c="1D460" xlink:href="#MJX-124-TEX-I-1D460"></use></g><g data-mml-node="mo" transform="translate(2381.8,0)"><use data-c="3D" xlink:href="#MJX-124-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(3437.6,0)"><g data-mml-node="mn" transform="translate(220,676)"><use data-c="31" xlink:href="#MJX-124-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-686)"><use data-c="32" xlink:href="#MJX-124-TEX-N-32"></use></g><rect width="700" height="60" x="120" y="220"></rect></g><g data-mml-node="munder" transform="translate(4544.2,0)"><g data-mml-node="mo"><use data-c="2211" xlink:href="#MJX-124-TEX-LO-2211"></use></g><g data-mml-node="mi" transform="translate(537.8,-1107.7) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-124-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(5988.2,0)"><use data-c="28" xlink:href="#MJX-124-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(6377.2,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-124-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-124-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(7540.8,0)"><use data-c="2212" xlink:href="#MJX-124-TEX-N-2212"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(8541.1,0)"><g data-mml-node="mover"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-124-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-124-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(470.7,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-124-TEX-N-5E"></use></g></g></g><g data-mml-node="msup" transform="translate(9482.5,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-124-TEX-N-29"></use></g><g data-mml-node="mn" transform="translate(422,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-124-TEX-N-32"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>L</mi><mi>o</mi><mi>s</mi><mi>s</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><munder><mo data-mjx-texclass="OP">∑</mo><mi>k</mi></munder><mo stretchy="false">(</mo><msub><mi>y</mi><mi>k</mi></msub><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math></mjx-assistive-mml></mjx-container></div></div><p><span>其中</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.13ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 941.4 647" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-73-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-73-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-73-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-73-TEX-I-1D458"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mi>k</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">y_k</script><span>为第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 521 705" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-85-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-85-TEX-I-1D458"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">k</script><span>个输出层节点的目标值（真实值），</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.13ex" height="2.296ex" role="img" focusable="false" viewBox="0 -810 941.4 1015" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-75-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-75-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-75-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-75-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-75-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(470.7,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-75-TEX-N-5E"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\hat{y_k}</script><span>为第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 521 705" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-85-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-85-TEX-I-1D458"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">k</script><span>个输出层节点的值（预测值）。</span></p><p><span>本项目中的代码实现如下：</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="C++"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="c++"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.5px; left: 8px;"><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: 0px; 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><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: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable-3">double</span> <span class="cm-variable">loss</span> <span class="cm-operator">=</span> <span class="cm-number">0.f</span>;</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">for</span> (<span class="cm-variable-3">size_t</span> <span class="cm-variable">k</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">k</span> <span class="cm-operator"><</span> <span class="cm-variable">Config::OUTNODE</span>; <span class="cm-operator">++</span><span class="cm-variable">k</span>) {</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable-3">double</span> <span class="cm-variable">tmp</span> <span class="cm-operator">=</span> <span class="cm-variable">std::fabs</span>(<span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span> <span class="cm-operator">-</span> <span class="cm-variable">label</span>[<span class="cm-variable">k</span>]);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">los</span> <span class="cm-operator">+=</span> <span class="cm-variable">tmp</span> <span class="cm-operator">*</span> <span class="cm-variable">tmp</span> <span class="cm-operator">/</span> <span class="cm-number">2</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">}</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 138px;"></div><div class="CodeMirror-gutters" style="display: none; height: 138px;"></div></div></div></pre><h3 id='3-backward反向传播）'><span>3. Backward（反向传播）</span></h3><p><span>利用梯度下降法进行优化。</span></p><h4 id='31-计算\nδλk\n输出层节点偏置值的修正值）'><span>3.1 计算</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.225ex" height="1.977ex" role="img" focusable="false" viewBox="0 -716 1867.4 873.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-77-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-77-TEX-I-1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path id="MJX-77-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-77-TEX-N-394"></use></g><g data-mml-node="msub" transform="translate(833,0)"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-77-TEX-I-1D706"></use></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-77-TEX-I-1D458"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><msub><mi>λ</mi><mi>k</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Delta \lambda_k</script><span>（输出层节点偏置值的修正值）</span></h4><p><span>其计算公式如下（激活函数为Sigmoid时）：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n475" cid="n475" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="28.829ex" height="2.398ex" role="img" focusable="false" viewBox="0 -810 12742.5 1060" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-125-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-125-TEX-I-1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path id="MJX-125-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-125-TEX-N-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 id="MJX-125-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-125-TEX-I-1D702" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-125-TEX-N-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 id="MJX-125-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-125-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path><path id="MJX-125-TEX-N-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 id="MJX-125-TEX-N-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="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-125-TEX-N-394"></use></g><g data-mml-node="msub" transform="translate(833,0)"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-125-TEX-I-1D706"></use></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-125-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(2145.2,0)"><use data-c="3D" xlink:href="#MJX-125-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(3201,0)"><use data-c="2212" xlink:href="#MJX-125-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(3979,0)"><use data-c="1D702" xlink:href="#MJX-125-TEX-I-1D702"></use></g><g data-mml-node="mo" transform="translate(4476,0)"><use data-c="28" xlink:href="#MJX-125-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(4865,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-125-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-125-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(6028.6,0)"><use data-c="2212" xlink:href="#MJX-125-TEX-N-2212"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(7028.8,0)"><g data-mml-node="mover"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-125-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-125-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(470.7,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-125-TEX-N-5E"></use></g></g></g><g data-mml-node="mo" transform="translate(7970.2,0)"><use data-c="29" xlink:href="#MJX-125-TEX-N-29"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(8359.2,0)"><g data-mml-node="mover"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-125-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-125-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(470.7,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-125-TEX-N-5E"></use></g></g></g><g data-mml-node="mo" transform="translate(9300.6,0)"><use data-c="28" xlink:href="#MJX-125-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(9689.6,0)"><use data-c="31" xlink:href="#MJX-125-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(10411.8,0)"><use data-c="2212" xlink:href="#MJX-125-TEX-N-2212"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(11412.1,0)"><g data-mml-node="mover"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-125-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-125-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(470.7,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-125-TEX-N-5E"></use></g></g></g><g data-mml-node="mo" transform="translate(12353.5,0)"><use data-c="29" xlink:href="#MJX-125-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="normal">Δ</mi><msub><mi>λ</mi><mi>k</mi></msub><mo>=</mo><mo>−</mo><mi>η</mi><mo stretchy="false">(</mo><msub><mi>y</mi><mi>k</mi></msub><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>其中</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.124ex" height="1.489ex" role="img" focusable="false" viewBox="0 -442 497 658" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.489ex;"><defs><path id="MJX-78-TEX-I-1D702" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D702" xlink:href="#MJX-78-TEX-I-1D702"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\eta</script><span>为学习率（其余变量上方已出现过不再进行标注）。</span></p><p><span>本项目中的代码实现如下：</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="C++"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="c++"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.5px; left: 8px;"><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: 0px; 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><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: 0px; width: 0px;"></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-3">size_t</span> <span class="cm-variable">k</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">k</span> <span class="cm-operator"><</span> <span class="cm-variable">Config::OUTNODE</span>; <span class="cm-operator">++</span><span class="cm-variable">k</span>) {</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable-3">double</span> <span class="cm-variable">bias_delta</span> <span class="cm-operator">=</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-operator">-</span>(<span class="cm-variable">label</span>[<span class="cm-variable">k</span>] <span class="cm-operator">-</span> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span>) <span class="cm-operator">*</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span> <span class="cm-operator">*</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> (<span class="cm-number">1.0</span> <span class="cm-operator">-</span> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">bias_delta</span> <span class="cm-operator">+=</span> <span class="cm-variable">bias_delta</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">}</span></pre></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="display: none; height: 184px;"></div></div></div></pre><h4 id='32-计算\nδvjk\n隐藏层节点到输出层节点权重的修正值）'><span>3.2 计算</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.662ex" height="2.286ex" role="img" focusable="false" viewBox="0 -716 2060.7 1010.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.666ex;"><defs><path id="MJX-79-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-79-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path id="MJX-79-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path><path id="MJX-79-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-79-TEX-N-394"></use></g><g data-mml-node="msub" transform="translate(833,0)"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-79-TEX-I-1D463"></use></g><g data-mml-node="TeXAtom" transform="translate(518,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D457" xlink:href="#MJX-79-TEX-I-1D457"></use></g><g data-mml-node="mi" transform="translate(412,0)"><use data-c="1D458" xlink:href="#MJX-79-TEX-I-1D458"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><msub><mi>v</mi><mrow data-mjx-texclass="ORD"><mi>j</mi><mi>k</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Delta v_{jk}</script><span>（隐藏层节点到输出层节点权重的修正值）</span></h4><p><span>其计算公式如下（激活函数为Sigmoid时）：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n481" cid="n481" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg 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106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-126-TEX-N-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 id="MJX-126-TEX-I-1D702" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 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xlink:href="#MJX-126-TEX-N-2212"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(6444.1,0)"><g data-mml-node="mover"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-126-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-126-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(470.7,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-126-TEX-N-5E"></use></g></g></g><g data-mml-node="mo" transform="translate(7385.5,0)"><use data-c="29" xlink:href="#MJX-126-TEX-N-29"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(7774.5,0)"><g data-mml-node="mover"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-126-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-126-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(470.7,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-126-TEX-N-5E"></use></g></g></g><g data-mml-node="mo" transform="translate(8715.9,0)"><use data-c="28" xlink:href="#MJX-126-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(9104.9,0)"><use data-c="31" xlink:href="#MJX-126-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(9827.2,0)"><use data-c="2212" xlink:href="#MJX-126-TEX-N-2212"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(10827.4,0)"><g data-mml-node="mover"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-126-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-126-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(470.7,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-126-TEX-N-5E"></use></g></g></g><g data-mml-node="mo" transform="translate(11768.8,0)"><use data-c="29" xlink:href="#MJX-126-TEX-N-29"></use></g><g data-mml-node="msub" transform="translate(12157.8,0)"><g data-mml-node="mi"><use data-c="210E" xlink:href="#MJX-126-TEX-I-210E"></use></g><g data-mml-node="mi" transform="translate(609,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-126-TEX-I-1D457"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="normal">Δ</mi><msub><mi>v</mi><mrow data-mjx-texclass="ORD"><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><mi>η</mi><mo stretchy="false">(</mo><msub><mi>y</mi><mi>k</mi></msub><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><msub><mi>h</mi><mi>j</mi></msub></math></mjx-assistive-mml></mjx-container></div></div><p><span>其中</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.15ex" height="2.236ex" role="img" focusable="false" viewBox="0 -694 950.3 988.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.666ex;"><defs><path id="MJX-80-TEX-I-210E" 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 id="MJX-80-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="210E" xlink:href="#MJX-80-TEX-I-210E"></use></g><g data-mml-node="mi" transform="translate(609,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-80-TEX-I-1D457"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>j</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">h_j</script><span>为第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.932ex" height="1.957ex" role="img" focusable="false" viewBox="0 -661 412 865" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.462ex;"><defs><path id="MJX-84-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D457" xlink:href="#MJX-84-TEX-I-1D457"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">j</script><span>个隐藏层节点的值（其余变量上方已出现过不再进行标注）。</span></p><p><span>本项目中的代码实现如下：</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="C++"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="c++"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.5px; left: 8px;"><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: 0px; 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><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: 0px; width: 0px;"></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-3">size_t</span> <span class="cm-variable">j</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">j</span> <span class="cm-operator"><</span> <span class="cm-variable">Config::HIDENODE</span>; <span class="cm-operator">++</span><span class="cm-variable">j</span>) {</span></pre></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-3">size_t</span> <span class="cm-variable">k</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">k</span> <span class="cm-operator"><</span> <span class="cm-variable">Config::OUTNODE</span>; <span class="cm-operator">++</span><span class="cm-variable">k</span>) {</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable-3">double</span> <span class="cm-variable">weight_delta</span> <span class="cm-operator">=</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> (<span class="cm-variable">label</span>[<span class="cm-variable">k</span>] <span class="cm-operator">-</span> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span>) <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span> <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> (<span class="cm-number">1.0</span> <span class="cm-operator">-</span> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span>) <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">hiddenLayer</span>[<span class="cm-variable">j</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-variable">hiddenLayer</span>[<span class="cm-variable">j</span>]<span class="cm-operator">-></span><span class="cm-variable">weight_delta</span>[<span class="cm-variable">k</span>] <span class="cm-operator">+=</span> <span class="cm-variable">weight_delta</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> }</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">}</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 253px;"></div><div class="CodeMirror-gutters" style="display: none; height: 253px;"></div></div></div></pre><h4 id='33-计算\nδβj\n隐藏层节点偏置值的修正值）'><span>3.3 计算</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.012ex" height="2.286ex" role="img" focusable="false" viewBox="0 -716 1773.3 1010.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.666ex;"><defs><path id="MJX-82-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-82-TEX-I-1D6FD" d="M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z"></path><path id="MJX-82-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-82-TEX-N-394"></use></g><g data-mml-node="msub" transform="translate(833,0)"><g data-mml-node="mi"><use data-c="1D6FD" xlink:href="#MJX-82-TEX-I-1D6FD"></use></g><g data-mml-node="mi" transform="translate(599,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-82-TEX-I-1D457"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><msub><mi>β</mi><mi>j</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Delta \beta_j</script><span>（隐藏层节点偏置值的修正值）</span></h4><p><span>其计算公式如下（激活函数为Sigmoid时）：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n487" cid="n487" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" 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transform="translate(412,0)"><use data-c="1D458" xlink:href="#MJX-127-TEX-I-1D458"></use></g></g></g><g data-mml-node="msub" transform="translate(15486.8,0)"><g data-mml-node="mi"><use data-c="210E" xlink:href="#MJX-127-TEX-I-210E"></use></g><g data-mml-node="mi" transform="translate(609,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-127-TEX-I-1D457"></use></g></g><g data-mml-node="mo" transform="translate(16437.1,0)"><use data-c="28" xlink:href="#MJX-127-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(16826.1,0)"><use data-c="31" xlink:href="#MJX-127-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(17548.3,0)"><use data-c="2212" xlink:href="#MJX-127-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(18548.6,0)"><g data-mml-node="mi"><use data-c="210E" xlink:href="#MJX-127-TEX-I-210E"></use></g><g data-mml-node="mi" transform="translate(609,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-127-TEX-I-1D457"></use></g></g><g data-mml-node="mo" transform="translate(19498.9,0)"><use data-c="29" xlink:href="#MJX-127-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="normal">Δ</mi><msub><mi>β</mi><mi>j</mi></msub><mo>=</mo><mo>−</mo><mi>η</mi><munder><mo data-mjx-texclass="OP">∑</mo><mi>k</mi></munder><mo stretchy="false">(</mo><msub><mi>y</mi><mi>k</mi></msub><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><msub><mi>v</mi><mrow data-mjx-texclass="ORD"><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>h</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>h</mi><mi>j</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>其中</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.778ex" height="1.668ex" role="img" focusable="false" viewBox="0 -443 1227.7 737.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.666ex;"><defs><path id="MJX-83-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 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data-c="1D458" xlink:href="#MJX-83-TEX-I-1D458"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mrow data-mjx-texclass="ORD"><mi>j</mi><mi>k</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">v_{jk}</script><span>为第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.932ex" height="1.957ex" role="img" focusable="false" viewBox="0 -661 412 865" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.462ex;"><defs><path id="MJX-84-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D457" xlink:href="#MJX-84-TEX-I-1D457"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">j</script><span>个隐藏层节点到第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 521 705" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-85-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-85-TEX-I-1D458"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">k</script><span>个输出层节点的权重（其余变量上方已出现过不再进行标注）。</span></p><p><span>本项目中的代码实现如下：</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="C++" style="break-inside: unset;"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="c++"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.5px; left: 8px;"><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: 0px; 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><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: 0px; width: 0px;"></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-3">size_t</span> <span class="cm-variable">j</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">j</span> <span class="cm-operator"><</span> <span class="cm-variable">Config::HIDENODE</span>; <span class="cm-operator">++</span><span class="cm-variable">j</span>) {</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-variable-3">double</span> <span class="cm-variable">bias_delta</span> <span class="cm-operator">=</span> <span class="cm-number">0.f</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-keyword">for</span> (<span class="cm-variable-3">size_t</span> <span class="cm-variable">k</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">k</span> <span class="cm-operator"><</span> <span class="cm-variable">Config::OUTNODE</span>; <span class="cm-operator">++</span><span class="cm-variable">k</span>) {</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-variable">bias_delta</span> <span class="cm-operator">+=</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-operator">-</span>(<span class="cm-variable">label</span>[<span class="cm-variable">k</span>] <span class="cm-operator">-</span> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span>) <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span> <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> (<span class="cm-number">1.0</span> <span class="cm-operator">-</span> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span>) <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">hiddenLayer</span>[<span class="cm-variable">j</span>]<span class="cm-operator">-></span><span class="cm-variable">weight</span>[<span class="cm-variable">k</span>];</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span>}</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-variable">bias_delta</span> <span class="cm-operator">*=</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">hiddenLayer</span>[<span class="cm-variable">j</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span> <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> (<span class="cm-number">1.0</span> <span class="cm-operator">-</span> <span class="cm-variable">hiddenLayer</span>[<span class="cm-variable">j</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-variable">hiddenLayer</span>[<span class="cm-variable">j</span>]<span class="cm-operator">-></span><span class="cm-variable">bias_delta</span> <span class="cm-operator">+=</span> <span class="cm-variable">bias_delta</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">}</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 345px;"></div><div class="CodeMirror-gutters" style="display: none; height: 345px;"></div></div></div></pre><h4 id='34-计算\nδwij\n输入层节点到隐藏层节点权重的修正值）'><span>3.4 计算</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.903ex" height="2.286ex" role="img" focusable="false" viewBox="0 -716 2167.3 1010.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.666ex;"><defs><path id="MJX-86-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-86-TEX-I-1D464" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 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display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><msub><mi>w</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Delta w_{ij}</script><span>（输入层节点到隐藏层节点权重的修正值）</span></h4><p><span>其计算公式如下（激活函数为Sigmoid时）：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n493" cid="n493" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="46.16ex" height="4.899ex" role="img" focusable="false" viewBox="0 -950 20402.8 2165.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.75ex;"><defs><path id="MJX-128-TEX-N-394" d="M51 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data-c="1D457" xlink:href="#MJX-128-TEX-I-1D457"></use></g></g><g data-mml-node="mo" transform="translate(19114.8,0)"><use data-c="29" xlink:href="#MJX-128-TEX-N-29"></use></g><g data-mml-node="msub" transform="translate(19503.8,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-128-TEX-I-1D465"></use></g><g data-mml-node="mi" transform="translate(605,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-128-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="normal">Δ</mi><msub><mi>w</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mi>η</mi><munder><mo data-mjx-texclass="OP">∑</mo><mi>k</mi></munder><mo stretchy="false">(</mo><msub><mi>y</mi><mi>k</mi></msub><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>y</mi><mi>k</mi></msub><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><msub><mi>v</mi><mrow data-mjx-texclass="ORD"><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>h</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>h</mi><mi>j</mi></msub><mo stretchy="false">)</mo><msub><mi>x</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container></div></div><p><span>其中</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.034ex" height="1.357ex" role="img" focusable="false" viewBox="0 -442 899 599.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-87-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 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-11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-87-TEX-I-1D465"></use></g><g data-mml-node="mi" transform="translate(605,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-87-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">x_i</script><span>为第</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.781ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 345 672" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-88-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-88-TEX-I-1D456"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">i</script><span>个输入层节点的值（其余变量上方已出现过不再进行标注）。</span></p><p><span>本项目中的代码实现如下：</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="C++" style="break-inside: unset;"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="c++"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.5px; left: 8px;"><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: 0px; 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><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: 0px; width: 0px;"></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-3">size_t</span> <span class="cm-variable">i</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">i</span> <span class="cm-operator"><</span> <span class="cm-variable">Config::INNODE</span>; <span class="cm-operator">++</span><span class="cm-variable">i</span>) {</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-keyword">for</span> (<span class="cm-variable-3">size_t</span> <span class="cm-variable">j</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">j</span> <span class="cm-operator"><</span> <span class="cm-variable">Config::HIDENODE</span>; <span class="cm-operator">++</span><span class="cm-variable">j</span>) {</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-variable-3">double</span> <span class="cm-variable">weight_delta</span> <span class="cm-operator">=</span> <span class="cm-number">0.f</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-keyword">for</span> (<span class="cm-variable-3">size_t</span> <span class="cm-variable">k</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">k</span> <span class="cm-operator"><</span> <span class="cm-variable">Config::OUTNODE</span>; <span class="cm-operator">++</span><span class="cm-variable">k</span>) {</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-variable">weight_delta</span> <span class="cm-operator">+=</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> (<span class="cm-variable">label</span>[<span class="cm-variable">k</span>] <span class="cm-operator">-</span> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span>) <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span> <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> (<span class="cm-number">1.0</span> <span class="cm-operator">-</span> <span class="cm-variable">outputLayer</span>[<span class="cm-variable">k</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span>) <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">hiddenLayer</span>[<span class="cm-variable">j</span>]<span class="cm-operator">-></span><span class="cm-variable">weight</span>[<span class="cm-variable">k</span>];</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> }</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-variable">weight_delta</span> <span class="cm-operator">*=</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">hiddenLayer</span>[<span class="cm-variable">j</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span> <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> (<span class="cm-number">1.0</span> <span class="cm-operator">-</span> <span class="cm-variable">hiddenLayer</span>[<span class="cm-variable">j</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span>) <span class="cm-operator">*</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">inputLayer</span>[<span class="cm-variable">i</span>]<span class="cm-operator">-></span><span class="cm-variable">value</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-variable">inputLayer</span>[<span class="cm-variable">i</span>]<span class="cm-operator">-></span><span class="cm-variable">weight_delta</span>[<span class="cm-variable">j</span>] <span class="cm-operator">+=</span> <span class="cm-variable">weight_delta</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span>}</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">}</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 414px;"></div><div class="CodeMirror-gutters" style="display: none; height: 414px;"></div></div></div></pre></div></div>
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