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<!DOCTYPE html><html lang="zh-CN" data-theme="light"><head><meta charset="UTF-8"><meta http-equiv="X-UA-Compatible" content="IE=edge"><meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no"><title>神经网络中的反向传播算法——BackPropagation算法 | BEIDAO.</title><meta name="keywords" content="深度学习,BP算法"><meta name="author" content="Beidaos"><meta name="copyright" content="Beidaos"><meta name="format-detection" content="telephone=no"><meta name="theme-color" content="#C6B3B1"><meta name="mobile-web-app-capable" content="yes"><meta name="apple-touch-fullscreen" content="yes"><meta name="apple-mobile-web-app-title" content="神经网络中的反向传播算法——BackPropagation算法"><meta name="application-name" content="神经网络中的反向传播算法——BackPropagation算法"><meta name="apple-mobile-web-app-capable" content="yes"><meta name="apple-mobile-web-app-status-bar-style" content="default"><link rel="bookmark" href="/img/siteicon/apple-touch-icon.png"><link rel="apple-touch-icon-precomposed" sizes="180x180" href="/img/siteicon/apple-touch-icon.png"><link rel="apple-touch-icon" sizes="192x192" href="/img/siteicon/apple-touch-icon.png"><link rel="apple-touch-icon" sizes="512x512" href="/img/siteicon/apple-touch-icon.png"><link rel="apple-touch-startup-image" media="screen and (device-width:320px) and (device-height:568px) and (-webkit-device-pixel-ratio:2) and (orientation:landscape)" href="/img/siteicon/splashIcons/icon_1136x640.png"><link rel="apple-touch-startup-image" media="screen and (device-width:320px) and (device-height:568px) and (-webkit-device-pixel-ratio:2) and (orientation:portrait)" href="/img/siteicon/splashIcons/icon_640x1136.png"><link rel="apple-touch-startup-image" media="screen and (device-width:414px) and (device-height:896px) and (-webkit-device-pixel-ratio:3) and (orientation:landscape)" href="/img/siteicon/splashIcons/icon_2688x1242.png"><link rel="apple-touch-startup-image" media="screen and (device-width:414px) and (device-height:896px) and (-webkit-device-pixel-ratio:2) and (orientation:landscape)" href="/img/siteicon/splashIcons/icon_1792x828.png"><link rel="apple-touch-startup-image" media="screen and (device-width:375px) and (device-height:812px) and (-webkit-device-pixel-ratio:3) and (orientation:portrait)" href="/img/siteicon/splashIcons/icon_1125x2436.png"><link rel="apple-touch-startup-image" media="screen and (device-width:414px) and (device-height:896px) and (-webkit-device-pixel-ratio:2) and (orientation:portrait)" href="/img/siteicon/splashIcons/icon_828x1792.png"><link rel="apple-touch-startup-image" media="screen and (device-width:375px) and (device-height:812px) and (-webkit-device-pixel-ratio:3) and (orientation:landscape)" href="/img/siteicon/splashIcons/icon_2436x1125.png"><link rel="apple-touch-startup-image" media="screen and (device-width:414px) and (device-height:736px) and (-webkit-device-pixel-ratio:3) and (orientation:portrait)" href="/img/siteicon/splashIcons/icon_1242x2208.png"><link rel="apple-touch-startup-image" media="screen and (device-width:414px) and (device-height:736px) and (-webkit-device-pixel-ratio:3) and (orientation:landscape)" href="/img/siteicon/splashIcons/icon_2208x1242.png"><link rel="apple-touch-startup-image" media="screen and (device-width:375px) and (device-height:667px) and (-webkit-device-pixel-ratio:2) and (orientation:landscape)" href="/img/siteicon/splashIcons/icon_1334x750.png"><link rel="apple-touch-startup-image" media="screen and (device-width:375px) and (device-height:667px) and (-webkit-device-pixel-ratio:2) and (orientation:portrait)" href="/img/siteicon/splashIcons/icon_750x1334.png"><link rel="apple-touch-startup-image" media="screen and (device-width:1024px) and (device-height:1366px) and (-webkit-device-pixel-ratio:2) and (orientation:landscape)" href="/img/siteicon/splashIcons/icon_2732x2048.png"><meta name="description" content="最近在看深度学习的东西，一开始看的吴恩达的UFLDL教程，有中文版就直接看了，后来发现有些地方总是不是很明确，又去看英文版，然后又找了些资料看，才发现，中文版的译者在翻译的时候会对省略的公式推导过程进行补充，但是补充的又是错的，难怪觉得有问题。反向传播法其实是神经网络的基础了，但是很多人在学的时候总是会遇到一些问题，或者看到大篇的公式觉得好像很难就退缩了，其实不难，就是一个链式求导法则反复用。如果">
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class="site-page" href="javascript:void(0);" rel="external nofollow noreferrer"><i class="anzhiyufont anzhiyu-icon-bars"></i></a></div></div></nav><div id="post-info"><div id="post-firstinfo"><div class="meta-firstline"><a class="post-meta-original">原创</a><span class="post-meta-categories"><span class="post-meta-separator"></span><i class="anzhiyufont anzhiyu-icon-inbox post-meta-icon"></i><a class="post-meta-categories" href="categories/%E6%B7%B1%E5%BA%A6%E5%AD%A6%E4%B9%A0/">深度学习</a></span><span class="article-meta tags"><a class="article-meta__tags" href="tags/%E6%B7%B1%E5%BA%A6%E5%AD%A6%E4%B9%A0/"><span><i class="iconfont icon-Tags"></i>深度学习</span></a><a class="article-meta__tags" href="tags/BP%E7%AE%97%E6%B3%95/"><span><i class="iconfont icon-Tags"></i>BP算法</span></a></span></div></div><h1 class="post-title">神经网络中的反向传播算法——BackPropagation算法</h1><div id="post-meta"><div class="meta-firstline"><i class="iconfont icon-zuozhe"></i><span class="post-meta-author">安志</span><span class="post-meta-date"><span class="post-meta-separator"></span><i class="anzhiyufont anzhiyu-icon-calendar-days post-meta-icon"></i><span class="post-meta-label">发表于</span><time class="post-meta-date-created" datetime="2023-07-25T10:00:00.000Z" title="发表于 2023-07-25 18:00:00">2023-07-25</time><span class="post-meta-separator"></span><i class="anzhiyufont anzhiyu-icon-history post-meta-icon"></i><span class="post-meta-label">更新于</span><time class="post-meta-date-updated" datetime="2024-03-01T07:11:02.449Z" title="更新于 2024-03-01 15:11:02">2024-03-01</time></span></div><div class="meta-secondline"><span class="post-meta-separator"></span><span class="post-meta-wordcount"><i class="anzhiyufont anzhiyu-icon-file-word post-meta-icon" title="文章字数"></i><span class="post-meta-label" title="文章字数">字数总计:</span><span class="word-count" title="文章字数">2.5k</span><span class="post-meta-separator"></span><i class="anzhiyufont anzhiyu-icon-clock post-meta-icon" title="阅读时长"></i><span class="post-meta-label" title="阅读时长">阅读时长:</span><span>10分钟</span></span><span class="post-meta-separator"></span><span class="post-meta-position" title="作者IP属地为北京"><i class="anzhiyufont anzhiyu-icon-location-dot"></i>北京</span></div></div></div><section class="main-hero-waves-area waves-area"><svg class="waves-svg" xmlns="http://www.w3.org/2000/svg" xlink="http://www.w3.org/1999/xlink" viewBox="0 24 150 28" preserveAspectRatio="none" shape-rendering="auto"><defs><path id="gentle-wave" d="M -160 44 c 30 0 58 -18 88 -18 s 58 18 88 18 s 58 -18 88 -18 s 58 18 88 18 v 44 h -352 Z"></path></defs><g class="parallax"><use href="#gentle-wave" x="48" y="0"></use><use href="#gentle-wave" x="48" y="3"></use><use href="#gentle-wave" x="48" y="5"></use><use href="#gentle-wave" x="48" y="7"></use></g></svg></section><div id="post-top-cover"><img class="nolazyload" id="post-top-bg" src="https://cdn.statically.io/gh/AnZhiJJ/Blog_Img@Science+/read_paper/1.kupf912yxi8.webp"></div></header><main class="layout" id="content-inner"><div id="post"><article class="post-content" id="article-container"><link rel="stylesheet external nofollow noreferrer" type="text/css" href="https://cdn.jsdelivr.net/hint.css/2.4.1/hint.min.css"><p>最近在看深度学习的东西，一开始看的吴恩达的UFLDL教程，有中文版就直接看了，后来发现有些地方总是不是很明确，又去看英文版，然后又找了些资料看，才发现，中文版的译者在翻译的时候会对省略的公式推导过程进行补充，但是补充的又是错的，难怪觉得有问题。反向传播法其实是神经网络的基础了，但是很多人在学的时候总是会遇到一些问题，或者看到大篇的公式觉得好像很难就退缩了，其实不难，就是一个链式求导法则反复用。如果不想看公式，可以直接把数值带进去，实际的计算一下，体会一下这个过程之后再来推导公式，这样就会觉得很容易了。</p>
<p>说到神经网络，大家看到这个图应该不陌生：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630140644406-409859737.png" alt="img"></p>
<p>这是典型的三层神经网络的基本构成，Layer L1是输入层，Layer L2是隐含层，Layer L3是隐含层，我们现在手里有一堆数据{x1,x2,x3,…,xn},输出也是一堆数据{y1,y2,y3,…,yn},现在要他们在隐含层做某种变换，让你把数据灌进去后得到你期望的输出。如果你希望你的输出和原始输入一样，那么就是最常见的自编码模型（Auto-Encoder）。可能有人会问，为什么要输入输出都一样呢？有什么用啊？其实应用挺广的，在图像识别，文本分类等等都会用到，我会专门再写一篇Auto-Encoder的文章来说明，包括一些变种之类的。如果你的输出和原始输入不一样，那么就是很常见的人工神经网络了，相当于让原始数据通过一个映射来得到我们想要的输出数据，也就是我们今天要讲的话题。</p>
<p>本文直接举一个例子，带入数值演示反向传播法的过程，公式的推导等到下次写Auto-Encoder的时候再写，其实也很简单，感兴趣的同学可以自己推导下试试：）（注：本文假设你已经懂得基本的神经网络构成，如果完全不懂，可以参考Poll写的笔记<sup id="fnref:1"><a href="#fn:1" rel="footnote"><span class="hint--top hint--error hint--medium hint--rounded hint--bounce" aria-label="Poll的笔记：[[Mechine Learning & Algorithm\] 神经网络基础](http://www.cnblogs.com/maybe2030/p/5597716.html)（http://www.cnblogs.com/maybe2030/p/5597716.html#3457159 ）">1</span></a></sup>。</p>
<p>假设，你有这样一个网络层：</p>
<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630141449671-1058672778.png" alt="img" style="zoom:50%;" />
<p>第一层是输入层，包含两个神经元i1，i2，和截距项b1；第二层是隐含层，包含两个神经元h1,h2和截距项b2，第三层是输出o1,o2，每条线上标的wi是层与层之间连接的权重，激活函数我们默认为sigmoid函数。</p>
<p>现在对他们赋上初值，如下图：</p>
<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630142019140-402363317.png" alt="img" style="zoom: 50%;" />
<p>其中，输入数据 i1=0.05，i2=0.10;</p>
<p>输出数据 o1=0.01,o2=0.99;</p>
<p>初始权重 w1=0.15,w2=0.20,w3=0.25,w4=0.30;</p>
<p>w5=0.40,w6=0.45,w7=0.50,w8=0.55</p>
<p>目标：给出输入数据i1,i2(0.05和0.10)，使输出尽可能与原始输出o1,o2(0.01和0.99)接近。</p>
<h2 id="Step-1-前向传播"><strong>Step 1 前向传播</strong></h2>
<p><strong>1.输入层---->隐含层：</strong></p>
<p>计算<strong>神经元h1的输入</strong>加权和：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630142915359-294460310.png" alt="img"></p>
<p><strong>神经元h1的输出o1:</strong>(此处<strong>用到激活函数为sigmoid函数</strong>)：</p>
<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630150115390-1035378028.png" alt="img" style="zoom: 80%;" />
<p>同理，可计算出神经元h2的输出o2：</p>
<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630150244265-1128303244.png" alt="img" style="zoom:80%;" />
<p><strong>2.隐含层---->输出层：</strong></p>
<p>计算输出层神经元h1、h2的输出o1和o2的值：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630150517109-389457135.png" alt="img"></p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630150638390-1210364296.png" alt="img"></p>
<p>这样前向传播的过程就结束了，<font color='red'>我们得到输出值为[0.75136079 , 0.772928465]，与实际值[0.01 , 0.99]相差还很远，现在我们对误差进行反向传播，更新权值，重新计算输出。</font></p>
<h2 id="Step-2-反向传播"><strong>Step 2 反向传播</strong></h2>
<p>目的：修改权重和偏差loss</p>
<h3 id="1-计算总误差"><strong>1.计算总误差</strong></h3>
<p>总误差：(square error)</p>
<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630151201812-1014280864.png" alt="img" style="zoom:67%;" />
<p>但是有两个输出，所以分别计算o1和o2的误差，总误差为两者之和：</p>
<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630151457593-1250510503.png" alt="img" style="zoom:67%;" />
<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630151508999-1967746600.png" alt="img" style="zoom:67%;" />
<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630151516093-1257166735.png" alt="img" style="zoom:67%;" />
<h3 id="2-隐含层-输出层的权值更新："><strong>2.隐含层---->输出层的权值更新：</strong></h3>
<p>以权重参数w5为例，如果我们想知道<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi><mn>5</mn></mrow><annotation encoding="application/x-tex">w5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord">5</span></span></span></span>对整体误差产生了多少影响，可以用整体误差对w5求偏导求出：（链式法则）</p>
<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630151916796-1001638091.png" alt="img" style="zoom:60%;" />
<p>下面的图可以更直观的看清楚<strong>误差是怎样反向传播的</strong>：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630152018906-1524325812.png" alt="img"></p>
<p>现在我们来分别计算每个式子的值：</p>
<p>计算<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630152206781-7976168.png" alt="img" style="zoom: 40%;" />：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630152258437-1960839452.png" alt="img"></p>
<p>计算<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630152417109-711077078.png" alt="img" style="zoom: 67%;" />：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630152511937-1667481051.png" alt="img"></p>
<p>（这一步实际上就是对sigmoid函数求导，比较简单，可以自己推导一下）</p>
<p>计算<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630152625593-2083321635.png" alt="img" style="zoom:67%;" />：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630152658109-214239362.png" alt="img"></p>
<p>最后三者相乘：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630152811640-888140287.png" alt="img"></p>
<p>这样我们就<strong>计算出整体误差E(total)对<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>w</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">w_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>的偏导值</strong>。</p>
<p>回过头来再看看上面的公式，我们发现：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630153103187-515052589.png" alt="img"></p>
<p>为了表达方便，用<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630153202812-585186566.png" alt="img" style="zoom:50%;" />来表示输出层的误差：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630153251234-1144531293.png" alt="img"></p>
<p>因此，整体误差E(total)对w5的偏导公式可以写成：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630153405296-436656179.png" alt="img"></p>
<p>如果输出层误差计为负的话，也可以写成：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630153514734-1544628024.png" alt="img"></p>
<p>最后我们来<strong>更新w5的值</strong>：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630153614374-1624035276.png" alt="img"></p>
<p>（其中，<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630153700093-743859667.png" alt="img" style="zoom: 50%;" />是学习速率，这里我们取0.5)</p>
<p>同理，可更新w6,w7,w8:</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630153807624-1231975059.png" alt="img"></p>
<h3 id="3-隐含层-隐含层的权值更新："><strong>3.隐含层---->隐含层的权值更新：</strong></h3>
<p>方法其实与上面说的差不多，但是有个地方需要变一下，在上文计算总误差对w5的偏导时，是从out(o1)---->net(o1)---->w5,但是<font color='red'>在隐含层之间的权值更新时，是out(h1)---->net(h1)---->w1,而out(h1)会接受E(o1)和E(o2)两个地方传来的误差，所以这个地方两个都要计算。</font></p>
<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630154317562-311369571.png" alt="img" style="zoom:50%;" />
<p>计算<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630154712202-1906007645.png" alt="img" style="zoom:67%;" />：<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630154758531-934861299.png" alt="img"></p>
<p>先计算<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630154958296-1922097086.png" alt="img" style="zoom:67%;" />：<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630155015546-1106216279.png" alt="img" style="zoom:67%;" /></p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630155036406-964647962.png" alt="img"></p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630155117656-1905928379.png" alt="img"></p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630155158468-157032005.png" alt="img"></p>
<p>同理，计算出：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630155310937-2103938446.png" alt="img"></p>
<p>两者相加得到总值：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630155435218-396769942.png" alt="img"></p>
<p>再计算<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630155555562-1422254830.png" alt="img">：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630155628046-229505495.png" alt="img"></p>
<p>再计算<img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630155731421-239852713.png" alt="img">：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630155706437-964861747.png" alt="img"></p>
<p>最后，三者相乘：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630155827718-189457408.png" alt="img"></p>
<p><strong>为了简化公式，用sigma(h1)表示隐含层单元h1的误差：</strong></p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630160345281-679307550.png" alt="img"></p>
<p>最后，更新w1的权值：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630160523437-1906004593.png" alt="img"></p>
<p>同理，额可更新w2,w3,w4的权值：</p>
<p><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" data-lazy-src="https://images2015.cnblogs.com/blog/853467/201606/853467-20160630160603484-1471434475.png" alt="img"></p>
<p>这样误差反向传播法就完成了，最后我们再把更新的权值重新计算，不停地迭代，在这个例子中第一次迭代之后，总误差E(total)由0.298371109下降至0.291027924。迭代10000次后，总误差为0.000035085，输出为<a href="%E5%8E%9F%E8%BE%93%E5%85%A5%E4%B8%BA%5B0.01,0.99%5D">0.015912196,0.984065734</a>,证明效果还是不错的。</p>
<figure class="highlight python"><table><tr><td class="code"><pre><span class="line"><span class="comment">#coding:utf-8</span></span><br><span class="line"><span class="keyword">import</span> random</span><br><span class="line"><span class="keyword">import</span> math</span><br><span class="line"></span><br><span class="line"><span class="comment">#</span></span><br><span class="line"><span class="comment"># 参数解释：</span></span><br><span class="line"><span class="comment"># "pd_" ：偏导的前缀</span></span><br><span class="line"><span class="comment"># "d_" ：导数的前缀</span></span><br><span class="line"><span class="comment"># "w_ho" ：隐含层到输出层的权重系数索引</span></span><br><span class="line"><span class="comment"># "w_ih" ：输入层到隐含层的权重系数的索引</span></span><br><span class="line"></span><br><span class="line"><span class="keyword">class</span> <span class="title class_">NeuralNetwork</span>:</span><br><span class="line"> LEARNING_RATE = <span class="number">0.5</span></span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">__init__</span>(<span class="params">self, num_inputs, num_hidden, num_outputs, hidden_layer_weights = <span class="literal">None</span>, hidden_layer_bias = <span class="literal">None</span>, output_layer_weights = <span class="literal">None</span>, output_layer_bias = <span class="literal">None</span></span>):</span><br><span class="line"> self.num_inputs = num_inputs</span><br><span class="line"></span><br><span class="line"> self.hidden_layer = NeuronLayer(num_hidden, hidden_layer_bias)</span><br><span class="line"> self.output_layer = NeuronLayer(num_outputs, output_layer_bias)</span><br><span class="line"></span><br><span class="line"> self.init_weights_from_inputs_to_hidden_layer_neurons(hidden_layer_weights)</span><br><span class="line"> self.init_weights_from_hidden_layer_neurons_to_output_layer_neurons(output_layer_weights)</span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">init_weights_from_inputs_to_hidden_layer_neurons</span>(<span class="params">self, hidden_layer_weights</span>):</span><br><span class="line"> weight_num = <span class="number">0</span></span><br><span class="line"> <span class="keyword">for</span> h <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(self.hidden_layer.neurons)):</span><br><span class="line"> <span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(self.num_inputs):</span><br><span class="line"> <span class="keyword">if</span> <span class="keyword">not</span> hidden_layer_weights:</span><br><span class="line"> self.hidden_layer.neurons[h].weights.append(random.random())</span><br><span class="line"> <span class="keyword">else</span>:</span><br><span class="line"> self.hidden_layer.neurons[h].weights.append(hidden_layer_weights[weight_num])</span><br><span class="line"> weight_num += <span class="number">1</span></span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">init_weights_from_hidden_layer_neurons_to_output_layer_neurons</span>(<span class="params">self, output_layer_weights</span>):</span><br><span class="line"> weight_num = <span class="number">0</span></span><br><span class="line"> <span class="keyword">for</span> o <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(self.output_layer.neurons)):</span><br><span class="line"> <span class="keyword">for</span> h <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(self.hidden_layer.neurons)):</span><br><span class="line"> <span class="keyword">if</span> <span class="keyword">not</span> output_layer_weights:</span><br><span class="line"> self.output_layer.neurons[o].weights.append(random.random())</span><br><span class="line"> <span class="keyword">else</span>:</span><br><span class="line"> self.output_layer.neurons[o].weights.append(output_layer_weights[weight_num])</span><br><span class="line"> weight_num += <span class="number">1</span></span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">inspect</span>(<span class="params">self</span>):</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">'------'</span>)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">'* Inputs: {}'</span>.<span class="built_in">format</span>(self.num_inputs))</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">'------'</span>)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">'Hidden Layer'</span>)</span><br><span class="line"> self.hidden_layer.inspect()</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">'------'</span>)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">'* Output Layer'</span>)</span><br><span class="line"> self.output_layer.inspect()</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">'------'</span>)</span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">feed_forward</span>(<span class="params">self, inputs</span>):</span><br><span class="line"> hidden_layer_outputs = self.hidden_layer.feed_forward(inputs)</span><br><span class="line"> <span class="keyword">return</span> self.output_layer.feed_forward(hidden_layer_outputs)</span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">train</span>(<span class="params">self, training_inputs, training_outputs</span>):</span><br><span class="line"> self.feed_forward(training_inputs)</span><br><span class="line"></span><br><span class="line"> <span class="comment"># 1. 输出神经元的值</span></span><br><span class="line"> pd_errors_wrt_output_neuron_total_net_input = [<span class="number">0</span>] * <span class="built_in">len</span>(self.output_layer.neurons)</span><br><span class="line"> <span class="keyword">for</span> o <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(self.output_layer.neurons)):</span><br><span class="line"></span><br><span class="line"> <span class="comment"># ∂E/∂zⱼ</span></span><br><span class="line"> pd_errors_wrt_output_neuron_total_net_input[o] = self.output_layer.neurons[o].calculate_pd_error_wrt_total_net_input(training_outputs[o])</span><br><span class="line"></span><br><span class="line"> <span class="comment"># 2. 隐含层神经元的值</span></span><br><span class="line"> pd_errors_wrt_hidden_neuron_total_net_input = [<span class="number">0</span>] * <span class="built_in">len</span>(self.hidden_layer.neurons)</span><br><span class="line"> <span class="keyword">for</span> h <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(self.hidden_layer.neurons)):</span><br><span class="line"></span><br><span class="line"> <span class="comment"># dE/dyⱼ = Σ ∂E/∂zⱼ * ∂z/∂yⱼ = Σ ∂E/∂zⱼ * wᵢⱼ</span></span><br><span class="line"> d_error_wrt_hidden_neuron_output = <span class="number">0</span></span><br><span class="line"> <span class="keyword">for</span> o <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(self.output_layer.neurons)):</span><br><span class="line"> d_error_wrt_hidden_neuron_output += pd_errors_wrt_output_neuron_total_net_input[o] * self.output_layer.neurons[o].weights[h]</span><br><span class="line"></span><br><span class="line"> <span class="comment"># ∂E/∂zⱼ = dE/dyⱼ * ∂zⱼ/∂</span></span><br><span class="line"> pd_errors_wrt_hidden_neuron_total_net_input[h] = d_error_wrt_hidden_neuron_output * self.hidden_layer.neurons[h].calculate_pd_total_net_input_wrt_input()</span><br><span class="line"></span><br><span class="line"> <span class="comment"># 3. 更新输出层权重系数</span></span><br><span class="line"> <span class="keyword">for</span> o <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(self.output_layer.neurons)):</span><br><span class="line"> <span class="keyword">for</span> w_ho <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(self.output_layer.neurons[o].weights)):</span><br><span class="line"></span><br><span class="line"> <span class="comment"># ∂Eⱼ/∂wᵢⱼ = ∂E/∂zⱼ * ∂zⱼ/∂wᵢⱼ</span></span><br><span class="line"> pd_error_wrt_weight = pd_errors_wrt_output_neuron_total_net_input[o] * self.output_layer.neurons[o].calculate_pd_total_net_input_wrt_weight(w_ho)</span><br><span class="line"></span><br><span class="line"> <span class="comment"># Δw = α * ∂Eⱼ/∂wᵢ</span></span><br><span class="line"> self.output_layer.neurons[o].weights[w_ho] -= self.LEARNING_RATE * pd_error_wrt_weight</span><br><span class="line"></span><br><span class="line"> <span class="comment"># 4. 更新隐含层的权重系数</span></span><br><span class="line"> <span class="keyword">for</span> h <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(self.hidden_layer.neurons)):</span><br><span class="line"> <span class="keyword">for</span> w_ih <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(self.hidden_layer.neurons[h].weights)):</span><br><span class="line"></span><br><span class="line"> <span class="comment"># ∂Eⱼ/∂wᵢ = ∂E/∂zⱼ * ∂zⱼ/∂wᵢ</span></span><br><span class="line"> pd_error_wrt_weight = pd_errors_wrt_hidden_neuron_total_net_input[h] * self.hidden_layer.neurons[h].calculate_pd_total_net_input_wrt_weight(w_ih)</span><br><span class="line"></span><br><span class="line"> <span class="comment"># Δw = α * ∂Eⱼ/∂wᵢ</span></span><br><span class="line"> self.hidden_layer.neurons[h].weights[w_ih] -= self.LEARNING_RATE * pd_error_wrt_weight</span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">calculate_total_error</span>(<span class="params">self, training_sets</span>):</span><br><span class="line"> total_error = <span class="number">0</span></span><br><span class="line"> <span class="keyword">for</span> t <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(training_sets)):</span><br><span class="line"> training_inputs, training_outputs = training_sets[t]</span><br><span class="line"> self.feed_forward(training_inputs)</span><br><span class="line"> <span class="keyword">for</span> o <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(training_outputs)):</span><br><span class="line"> total_error += self.output_layer.neurons[o].calculate_error(training_outputs[o])</span><br><span class="line"> <span class="keyword">return</span> total_error</span><br><span class="line"></span><br><span class="line"><span class="keyword">class</span> <span class="title class_">NeuronLayer</span>:</span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">__init__</span>(<span class="params">self, num_neurons, bias</span>):</span><br><span class="line"></span><br><span class="line"> <span class="comment"># 同一层的神经元共享一个截距项b</span></span><br><span class="line"> self.bias = bias <span class="keyword">if</span> bias <span class="keyword">else</span> random.random()</span><br><span class="line"></span><br><span class="line"> self.neurons = []</span><br><span class="line"> <span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(num_neurons):</span><br><span class="line"> self.neurons.append(Neuron(self.bias))</span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">inspect</span>(<span class="params">self</span>):</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">'Neurons:'</span>, <span class="built_in">len</span>(self.neurons))</span><br><span class="line"> <span class="keyword">for</span> n <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(self.neurons)):</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">' Neuron'</span>, n)</span><br><span class="line"> <span class="keyword">for</span> w <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(self.neurons[n].weights)):</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">' Weight:'</span>, self.neurons[n].weights[w])</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">' Bias:'</span>, self.bias)</span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">feed_forward</span>(<span class="params">self, inputs</span>):</span><br><span class="line"> outputs = []</span><br><span class="line"> <span class="keyword">for</span> neuron <span class="keyword">in</span> self.neurons:</span><br><span class="line"> outputs.append(neuron.calculate_output(inputs))</span><br><span class="line"> <span class="keyword">return</span> outputs</span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">get_outputs</span>(<span class="params">self</span>):</span><br><span class="line"> outputs = []</span><br><span class="line"> <span class="keyword">for</span> neuron <span class="keyword">in</span> self.neurons:</span><br><span class="line"> outputs.append(neuron.output)</span><br><span class="line"> <span class="keyword">return</span> outputs</span><br><span class="line"></span><br><span class="line"><span class="keyword">class</span> <span class="title class_">Neuron</span>:</span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">__init__</span>(<span class="params">self, bias</span>):</span><br><span class="line"> self.bias = bias</span><br><span class="line"> self.weights = []</span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">calculate_output</span>(<span class="params">self, inputs</span>):</span><br><span class="line"> self.inputs = inputs</span><br><span class="line"> self.output = self.squash(self.calculate_total_net_input())</span><br><span class="line"> <span class="keyword">return</span> self.output</span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">calculate_total_net_input</span>(<span class="params">self</span>):</span><br><span class="line"> total = <span class="number">0</span></span><br><span class="line"> <span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(self.inputs)):</span><br><span class="line"> total += self.inputs[i] * self.weights[i]</span><br><span class="line"> <span class="keyword">return</span> total + self.bias</span><br><span class="line"></span><br><span class="line"> <span class="comment"># 激活函数sigmoid</span></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">squash</span>(<span class="params">self, total_net_input</span>):</span><br><span class="line"> <span class="keyword">return</span> <span class="number">1</span> / (<span class="number">1</span> + math.exp(-total_net_input))</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">calculate_pd_error_wrt_total_net_input</span>(<span class="params">self, target_output</span>):</span><br><span class="line"> <span class="keyword">return</span> self.calculate_pd_error_wrt_output(target_output) * self.calculate_pd_total_net_input_wrt_input();</span><br><span class="line"></span><br><span class="line"> <span class="comment"># 每一个神经元的误差是由平方差公式计算的</span></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">calculate_error</span>(<span class="params">self, target_output</span>):</span><br><span class="line"> <span class="keyword">return</span> <span class="number">0.5</span> * (target_output - self.output) ** <span class="number">2</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">calculate_pd_error_wrt_output</span>(<span class="params">self, target_output</span>):</span><br><span class="line"> <span class="keyword">return</span> -(target_output - self.output)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">calculate_pd_total_net_input_wrt_input</span>(<span class="params">self</span>):</span><br><span class="line"> <span class="keyword">return</span> self.output * (<span class="number">1</span> - self.output)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"> <span class="keyword">def</span> <span class="title function_">calculate_pd_total_net_input_wrt_weight</span>(<span class="params">self, index</span>):</span><br><span class="line"> <span class="keyword">return</span> self.inputs[index]</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="comment"># 文中的例子:</span></span><br><span class="line"></span><br><span class="line">nn = NeuralNetwork(<span class="number">2</span>, <span class="number">2</span>, <span class="number">2</span>, hidden_layer_weights=[<span class="number">0.15</span>, <span class="number">0.2</span>, <span class="number">0.25</span>, <span class="number">0.3</span>], hidden_layer_bias=<span class="number">0.35</span>, output_layer_weights=[<span class="number">0.4</span>, <span class="number">0.45</span>, <span class="number">0.5</span>, <span class="number">0.55</span>], output_layer_bias=<span class="number">0.6</span>)</span><br><span class="line"><span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(<span class="number">10000</span>):</span><br><span class="line"> nn.train([<span class="number">0.05</span>, <span class="number">0.1</span>], [<span class="number">0.01</span>, <span class="number">0.09</span>])</span><br><span class="line"> <span class="built_in">print</span>(i, <span class="built_in">round</span>(nn.calculate_total_error([[[<span class="number">0.05</span>, <span class="number">0.1</span>], [<span class="number">0.01</span>, <span class="number">0.09</span>]]]), <span class="number">9</span>))</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="comment">#另外一个例子，可以把上面的例子注释掉再运行一下:</span></span><br><span class="line"></span><br><span class="line"><span class="comment"># training_sets = [</span></span><br><span class="line"><span class="comment"># [[0, 0], [0]],</span></span><br><span class="line"><span class="comment"># [[0, 1], [1]],</span></span><br><span class="line"><span class="comment"># [[1, 0], [1]],</span></span><br><span class="line"><span class="comment"># [[1, 1], [0]]</span></span><br><span class="line"><span class="comment"># ]</span></span><br><span class="line"></span><br><span class="line"><span class="comment"># nn = NeuralNetwork(len(training_sets[0][0]), 5, len(training_sets[0][1]))</span></span><br><span class="line"><span class="comment"># for i in range(10000):</span></span><br><span class="line"><span class="comment"># training_inputs, training_outputs = random.choice(training_sets)</span></span><br><span class="line"><span class="comment"># nn.train(training_inputs, training_outputs)</span></span><br><span class="line"><span class="comment"># print(i, nn.calculate_total_error(training_sets))</span></span><br></pre></td></tr></table></figure>
<p> 稳重使用的是sigmoid激活函数，实际还有几种不同的激活函数可以选择，具体的可以参考文献<sup id="fnref:3"><a href="#fn:3" rel="footnote"><span class="hint--top hint--error hint--medium hint--rounded hint--bounce" aria-label="http://www.cedar.buffalo.edu/%7Esrihari/CSE574/Chap5/Chap5.3-BackProp.pdf">3</span></a></sup>，最后推荐一个在线演示神经网络变化的网址：<a target="_blank" rel="noopener external nofollow noreferrer" href="http://www.emergentmind.com/neural-network%EF%BC%8C%E5%8F%AF%E4%BB%A5%E8%87%AA%E5%B7%B1%E5%A1%AB%E8%BE%93%E5%85%A5%E8%BE%93%E5%87%BA%EF%BC%8C%E7%84%B6%E5%90%8E%E8%A7%82%E7%9C%8B%E6%AF%8F%E4%B8%80%E6%AC%A1%E8%BF%AD%E4%BB%A3%E6%9D%83%E5%80%BC%E7%9A%84%E5%8F%98%E5%8C%96%EF%BC%8C%E5%BE%88%E5%A5%BD%E7%8E%A9~%E5%A6%82%E6%9E%9C%E6%9C%89%E9%94%99%E8%AF%AF%E7%9A%84%E6%88%96%E8%80%85%E4%B8%8D%E6%87%82%E7%9A%84%E6%AC%A2%E8%BF%8E%E7%95%99%E8%A8%80%EF%BC%9A">http://www.emergentmind.com/neural-network，可以自己填输入输出，然后观看每一次迭代权值的变化，很好玩~如果有错误的或者不懂的欢迎留言：</a></p>
<p>参考文献：</p>
<p>2.Rachel_Zhang:<a target="_blank" rel="noopener external nofollow noreferrer" href="http://blog.csdn.net/abcjennifer/article/details/7758797">http://blog.csdn.net/abcjennifer/article/details/7758797</a></p>
<p>4.<a target="_blank" rel="noopener external nofollow noreferrer" href="https://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/">https://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/</a><div id="footnotes"><hr><div id="footnotelist"><ol style="list-style: none; padding-left: 0; margin-left: 40px"><li id="fn:1"><span style="display: inline-block; vertical-align: top; padding-right: 10px; margin-left: -40px">1.</span><span style="display: inline-block; vertical-align: top; margin-left: 10px;">Poll的笔记：[<a target="_blank" rel="noopener external nofollow noreferrer" href="http://www.cnblogs.com/maybe2030/p/5597716.html">Mechine Learning & Algorithm] 神经网络基础</a>（<a target="_blank" rel="noopener external nofollow noreferrer" href="http://www.cnblogs.com/maybe2030/p/5597716.html#3457159">http://www.cnblogs.com/maybe2030/p/5597716.html#3457159</a> ）<a href="#fnref:1" rev="footnote">↩</a></span></li><li id="fn:3"><span style="display: inline-block; vertical-align: top; padding-right: 10px; margin-left: -40px">3.</span><span style="display: inline-block; vertical-align: top; margin-left: 10px;"><a target="_blank" rel="noopener external nofollow noreferrer" href="http://www.cedar.buffalo.edu/%7Esrihari/CSE574/Chap5/Chap5.3-BackProp.pdf">http://www.cedar.buffalo.edu/~srihari/CSE574/Chap5/Chap5.3-BackProp.pdf</a><a href="#fnref:3" rev="footnote">↩</a></span></li></ol></div></div></p>
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!hoverOnCommentBarrage
) {
removeCommentBarrage(commentBarrageConfig.barrageTimer.shift());
}
}, commentBarrageConfig.barrageTime);
}
function commentLinkFilter(data) {
data.sort((a, b) => {
return a.created - b.created;
});
let newData = [];
data.forEach(item => {
newData.push(...getCommentReplies(item));
});
return newData;
}
function getCommentReplies(item) {
if (item.replies) {
let replies = [item];
item.replies.forEach(item => {
replies.push(...getCommentReplies(item));
});
return replies;
} else {
return [];
}
}
function popCommentBarrage(data) {
let barrage = document.createElement("div");
barrage.className = "comment-barrage-item";
barrage.innerHTML = `
<div class="barrageHead">
<a class="barrageTitle ${
data.mailMd5 === "d338f432ad0bf2f61e5fe4ad1642725d" ? "barrageBloggerTitle" : ""
}" href="javascript:anzhiyu.scrollTo('#post-comment')"">
${data.mailMd5 === "d338f432ad0bf2f61e5fe4ad1642725d" ? "博主" : "热评"}
</a>
<div class="barrageNick">${data.nick}</div>
<img class="nolazyload barrageAvatar" src="https://cravatar.cn/avatar/${data.mailMd5}"/>
<a class="comment-barrage-close" href="javascript:anzhiyu.switchCommentBarrage()" rel="external nofollow noreferrer"><i class="anzhiyufont anzhiyu-icon-xmark"></i></a>
</div>
<a class="barrageContent" href="#${data.id}">
<object>${data.comment}</object>
</a>
`;
commentBarrageConfig.barrageTimer.push(barrage);
commentBarrageConfig.dom.append(barrage);
}
function removeCommentBarrage(barrage) {
barrage.className = "comment-barrage-item out";
setTimeout(() => {
if (commentBarrageConfig.dom && commentBarrageConfig.dom.contains(barrage)) {
commentBarrageConfig.dom.removeChild(barrage);
}
}, 1000);
}
// 自动隐藏
const commentEntryCallback = (entries) => {
const commentBarrage = document.querySelector(".comment-barrage");
const postComment = document.getElementById("post-comment");
entries.forEach(entry => {
if (postComment && commentBarrage && document.body.clientWidth > 768) {
commentBarrage.style.bottom = entry.isIntersecting ? "-200px" : "0";
}
});
};
// 创建IntersectionObserver实例
const observer = new IntersectionObserver(commentEntryCallback, {
root: null,
rootMargin: "0px",
threshold: 0
});
// 监视目标元素
const postCommentTarget = document.getElementById("post-comment");
if (postCommentTarget) {
observer.observe(postCommentTarget);
}
initCommentBarrage();
if (localStorage.getItem("commentBarrageSwitch") !== "false") {
document.querySelector(".comment-barrage").style.display = "flex";
document.querySelector(".menu-commentBarrage-text").textContent = "关闭热评";
} else {
document.querySelector(".comment-barrage").style.display = "none";
document.querySelector(".menu-commentBarrage-text").textContent = "显示热评";
}
document.addEventListener("pjax:send", function () {
clearInterval(commentInterval);
});
}</script><script data-pjax src="https://npm.elemecdn.com/[email protected]/catalog-bar/catalog-bar.js"></script><script async data-pjax src="https://npm.elemecdn.com/[email protected]/categoryBar/categoryBar.js"></script><script async data-pjax src="https://npm.elemecdn.com/[email protected]/waterfall/waterfall.js"></script><script src="https://lf3-cdn-tos.bytecdntp.com/cdn/expire-1-M/qrcodejs/1.0.0/qrcode.min.js"></script><script>// 初始化函数
let rm = {};
//禁止图片拖拽
let imgElements = document.getElementsByTagName("img");
for (let i = 0; i < imgElements.length; i++) {
imgElements[i].addEventListener("dragstart", function (event) {
event.preventDefault();
});
}
// 显示菜单
rm.showRightMenu = function (isTrue, x = 0, y = 0) {
console.info(x, y)
let rightMenu = document.getElementById("rightMenu");
rightMenu.style.top = x + "px";
rightMenu.style.left = y + "px";
if (isTrue) {
rightMenu.style.display = "block";
stopMaskScroll();
} else {
rightMenu.style.display = "none";
}
};
// 隐藏菜单
rm.hideRightMenu = function () {
rm.showRightMenu(false);
let rightMenuMask = document.querySelector("#rightmenu-mask");
rightMenuMask.style.display = "none";
};
// 尺寸
let rmWidth = document.getElementById("rightMenu").offsetWidth;
let rmHeight = document.getElementById("rightMenu").offsetHeight;
// 重新定义尺寸
rm.reloadrmSize = function () {
rightMenu.style.visibility = "hidden";
rightMenu.style.display = "block";
// 获取宽度和高度
rmWidth = document.getElementById("rightMenu").offsetWidth;
rmHeight = document.getElementById("rightMenu").offsetHeight;
rightMenu.style.visibility = "visible";
};
// 获取点击的href
let domhref = "";
let domImgSrc = "";
let globalEvent = null;
var oncontextmenuFunction = function (event) {
if (document.body.clientWidth > 768) {
let pageX = event.clientX + 10; //加10是为了防止显示时鼠标遮在菜单上
let pageY = event.clientY;
//其他额外菜单
const $rightMenuOther = document.querySelector(".rightMenuOther");
const $rightMenuPlugin = document.querySelector(".rightMenuPlugin");
const $rightMenuCopyText = document.querySelector("#menu-copytext");
const $rightMenuPasteText = document.querySelector("#menu-pastetext");
const $rightMenuCommentText = document.querySelector("#menu-commenttext");
const $rightMenuNewWindow = document.querySelector("#menu-newwindow");
const $rightMenuNewWindowImg = document.querySelector("#menu-newwindowimg");
const $rightMenuCopyLink = document.querySelector("#menu-copylink");
const $rightMenuCopyImg = document.querySelector("#menu-copyimg");
const $rightMenuDownloadImg = document.querySelector("#menu-downloadimg");
const $rightMenuSearch = document.querySelector("#menu-search");
const $rightMenuSearchBaidu = document.querySelector("#menu-searchBaidu");
const $rightMenuMusicToggle = document.querySelector("#menu-music-toggle");
const $rightMenuMusicBack = document.querySelector("#menu-music-back");
const $rightMenuMusicForward = document.querySelector("#menu-music-forward");
const $rightMenuMusicPlaylist = document.querySelector("#menu-music-playlist");
const $rightMenuMusicCopyMusicName = document.querySelector("#menu-music-copyMusicName");
let href = event.target.href;
let imgsrc = event.target.currentSrc;
// 判断模式 扩展模式为有事件
let pluginMode = false;
$rightMenuOther.style.display = "block";
globalEvent = event;
// 检查是否需要复制 是否有选中文本
if (selectTextNow && window.getSelection()) {
pluginMode = true;
$rightMenuCopyText.style.display = "block";
$rightMenuCommentText.style.display = "block";
$rightMenuSearch.style.display = "block";
$rightMenuSearchBaidu.style.display = "block";
} else {
$rightMenuCopyText.style.display = "none";
$rightMenuCommentText.style.display = "none";
$rightMenuSearchBaidu.style.display = "none";
$rightMenuSearch.style.display = "none";
}
//检查是否右键点击了链接a标签
if (href) {
pluginMode = true;
$rightMenuNewWindow.style.display = "block";
$rightMenuCopyLink.style.display = "block";
domhref = href;
} else {
$rightMenuNewWindow.style.display = "none";
$rightMenuCopyLink.style.display = "none";
}
//检查是否需要复制图片
if (imgsrc) {
pluginMode = true;
$rightMenuCopyImg.style.display = "block";
$rightMenuDownloadImg.style.display = "block";
$rightMenuNewWindowImg.style.display = "block";
document.getElementById("rightMenu").style.width="12rem"
domImgSrc = imgsrc;
} else {
$rightMenuCopyImg.style.display = "none";
$rightMenuDownloadImg.style.display = "none";
$rightMenuNewWindowImg.style.display = "none";
}
// 判断是否为输入框
if (event.target.tagName.toLowerCase() === "input" || event.target.tagName.toLowerCase() === "textarea") {
pluginMode = true;
$rightMenuPasteText.style.display = "block";
} else {
$rightMenuPasteText.style.display = "none";
}
const navMusicEl = document.querySelector("#nav-music");
//判断是否是音乐
if (navMusicEl && navMusicEl.contains(event.target)) {
pluginMode = true;
$rightMenuMusicToggle.style.display = "block";
$rightMenuMusicBack.style.display = "block";
$rightMenuMusicForward.style.display = "block";
$rightMenuMusicPlaylist.style.display = "block";
$rightMenuMusicCopyMusicName.style.display = "block";
} else {
$rightMenuMusicToggle.style.display = "none";
$rightMenuMusicBack.style.display = "none";
$rightMenuMusicForward.style.display = "none";
$rightMenuMusicPlaylist.style.display = "none";
$rightMenuMusicCopyMusicName.style.display = "none";
}
// 如果不是扩展模式则隐藏扩展模块
if (pluginMode) {
$rightMenuOther.style.display = "none";
$rightMenuPlugin.style.display = "block";
} else {
$rightMenuPlugin.style.display = "none";
}
rm.reloadrmSize();
// 鼠标默认显示在鼠标右下方，当鼠标靠右或靠下时，将菜单显示在鼠标左方\上方
if (pageX + rmWidth > window.innerWidth) {
pageX -= rmWidth + 10;
}
if (pageY + rmHeight > window.innerHeight) {
pageY -= pageY + rmHeight - window.innerHeight;
}
rm.showRightMenu(true, pageY, pageX);
document.getElementById("rightmenu-mask").style.display = "flex";
return false;
}
};
// 监听右键初始化
window.oncontextmenu = oncontextmenuFunction
// 下载图片状态
rm.downloadimging = false;
// 复制图片到剪贴板
rm.writeClipImg = function (imgsrc) {
console.log("按下复制");
rm.hideRightMenu();
anzhiyu.snackbarShow("正在下载中，请稍后", false, 10000);
if (rm.downloadimging == false) {
rm.downloadimging = true;
setTimeout(function () {
copyImage(imgsrc);
anzhiyu.snackbarShow("复制成功！图片已添加盲水印，请遵守版权协议");
rm.downloadimging = false;
}, "10000");
}
};
function imageToBlob(imageURL) {
const img = new Image();
const c = document.createElement("canvas");
const ctx = c.getContext("2d");
img.crossOrigin = "";
img.src = imageURL;
return new Promise(resolve => {
img.onload = function () {
c.width = this.naturalWidth;
c.height = this.naturalHeight;
ctx.drawImage(this, 0, 0);
c.toBlob(
blob => {
// here the image is a blob
resolve(blob);
},
"image/png",
0.75
);
};
});
}
async function copyImage(imageURL) {
const blob = await imageToBlob(imageURL);
const item = new ClipboardItem({ "image/png": blob });
navigator.clipboard.write([item]);
}
rm.switchDarkMode = function () {
// Switch Between Light And Dark Mode
const nowMode = document.documentElement.getAttribute("data-theme") === "dark" ? "dark" : "light";
if (nowMode === "light") {
activateDarkMode();
saveToLocal.set("theme", "dark", 2);
GLOBAL_CONFIG.Snackbar !== undefined && anzhiyu.snackbarShow(GLOBAL_CONFIG.Snackbar.day_to_night);
document.querySelector(".menu-darkmode-text").textContent = "浅色模式";
} else {
activateLightMode();
saveToLocal.set("theme", "light", 2);
GLOBAL_CONFIG.Snackbar !== undefined && anzhiyu.snackbarShow(GLOBAL_CONFIG.Snackbar.night_to_day);