register_grad_op.html



<!DOCTYPE html>
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" > <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" > <!--<![endif]-->
<head>
  <meta charset="utf-8">
  
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  
  <title>Design Doc: Gradient Operators Registration &mdash; PaddlePaddle  文档</title>
  

    <link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
  

        <link rel="index" title="索引"
              href="../genindex.html"/>
        <link rel="search" title="搜索" href="../search.html"/>
    <link rel="top" title="PaddlePaddle  文档" href="../index.html"/> 

  <link rel="stylesheet" href="https://cdn.jsdelivr.net/perfect-scrollbar/0.6.14/css/perfect-scrollbar.min.css" type="text/css" />
  <link rel="stylesheet" href="../_static/css/override.css" type="text/css" />
  <script>
  var _hmt = _hmt || [];
  (function() {
    var hm = document.createElement("script");
    hm.src = "//hm.baidu.com/hm.js?b9a314ab40d04d805655aab1deee08ba";
    var s = document.getElementsByTagName("script")[0]; 
    s.parentNode.insertBefore(hm, s);
  })();
  </script>

  
  <script src="../_static/js/modernizr.min.js"></script>

</head>

<body class="wy-body-for-nav" role="document">

  
  <header class="site-header">
    <div class="site-logo">
      <a href="/"><img src="../_static/images/PP_w.png"></a>
    </div>
    <div class="site-nav-links">
      <div class="site-menu">
        <a class="fork-on-github" href="https://github.com/PaddlePaddle/Paddle" target="_blank"><i class="fa fa-github"></i>Fork me on Github</a>
        <div class="language-switcher dropdown">
          <a type="button" data-toggle="dropdown">
            <span>English</span>
            <i class="fa fa-angle-up"></i>
            <i class="fa fa-angle-down"></i>
          </a>
          <ul class="dropdown-menu">
            <li><a href="/doc_cn">中文</a></li>
            <li><a href="/doc">English</a></li>
          </ul>
        </div>
        <ul class="site-page-links">
          <li><a href="/">Home</a></li>
        </ul>
      </div>
      <div class="doc-module">
        
        <ul>
<li class="toctree-l1"><a class="reference internal" href="../getstarted/index_cn.html">新手入门</a></li>
<li class="toctree-l1"><a class="reference internal" href="../howto/index_cn.html">进阶指南</a></li>
<li class="toctree-l1"><a class="reference internal" href="../api/index_cn.html">API</a></li>
<li class="toctree-l1"><a class="reference internal" href="../faq/index_cn.html">FAQ</a></li>
<li class="toctree-l1"><a class="reference internal" href="../mobile/index_cn.html">MOBILE</a></li>
</ul>

        
<div role="search">
  <form id="rtd-search-form" class="wy-form" action="../search.html" method="get">
    <input type="text" name="q" placeholder="Search docs" />
    <input type="hidden" name="check_keywords" value="yes" />
    <input type="hidden" name="area" value="default" />
  </form>
</div>        
      </div>
    </div>
  </header>
  
  <div class="main-content-wrap">

    
    <nav class="doc-menu-vertical" role="navigation">
        
          
          <ul>
<li class="toctree-l1"><a class="reference internal" href="../getstarted/index_cn.html">新手入门</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../getstarted/build_and_install/index_cn.html">安装与编译</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../getstarted/build_and_install/pip_install_cn.html">使用pip安装</a></li>
<li class="toctree-l3"><a class="reference internal" href="../getstarted/build_and_install/docker_install_cn.html">使用Docker安装运行</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/dev/build_cn.html">用Docker编译和测试PaddlePaddle</a></li>
<li class="toctree-l3"><a class="reference internal" href="../getstarted/build_and_install/build_from_source_cn.html">从源码编译</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="../getstarted/concepts/use_concepts_cn.html">基本使用概念</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../howto/index_cn.html">进阶指南</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../howto/usage/cmd_parameter/index_cn.html">设置命令行参数</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../howto/usage/cmd_parameter/use_case_cn.html">使用案例</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/usage/cmd_parameter/arguments_cn.html">参数概述</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/usage/cmd_parameter/detail_introduction_cn.html">细节描述</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="../howto/usage/cluster/cluster_train_cn.html">PaddlePaddle分布式训练</a></li>
<li class="toctree-l2"><a class="reference internal" href="../howto/usage/k8s/k8s_basis_cn.html">Kubernetes 简介</a></li>
<li class="toctree-l2"><a class="reference internal" href="../howto/usage/k8s/k8s_cn.html">Kubernetes单机训练</a></li>
<li class="toctree-l2"><a class="reference internal" href="../howto/usage/k8s/k8s_distributed_cn.html">Kubernetes分布式训练</a></li>
<li class="toctree-l2"><a class="reference internal" href="../howto/dev/contribute_to_paddle_cn.html">如何贡献代码</a></li>
<li class="toctree-l2"><a class="reference internal" href="../howto/dev/write_docs_cn.html">如何贡献/修改文档</a></li>
<li class="toctree-l2"><a class="reference internal" href="../howto/deep_model/rnn/index_cn.html">RNN相关模型</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../howto/deep_model/rnn/rnn_config_cn.html">RNN配置</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/deep_model/rnn/recurrent_group_cn.html">Recurrent Group教程</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/deep_model/rnn/hierarchical_layer_cn.html">支持双层序列作为输入的Layer</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/deep_model/rnn/hrnn_rnn_api_compare_cn.html">单双层RNN API对比介绍</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="../howto/optimization/gpu_profiling_cn.html">GPU性能分析与调优</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../api/index_cn.html">API</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../api/v2/model_configs.html">模型配置</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/config/activation.html">Activation</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/config/layer.html">Layers</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/config/evaluators.html">Evaluators</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/config/optimizer.html">Optimizer</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/config/pooling.html">Pooling</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/config/networks.html">Networks</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/config/attr.html">Parameter Attribute</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="../api/v2/data.html">数据访问</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/data/data_reader.html">Data Reader Interface</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/data/image.html">Image Interface</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/data/dataset.html">Dataset</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="../api/v2/run_logic.html">训练与应用</a></li>
<li class="toctree-l2"><a class="reference internal" href="../api/v2/fluid.html">Fluid</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/fluid/layers.html">Layers</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/fluid/data_feeder.html">DataFeeder</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/fluid/executor.html">Executor</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/fluid/initializer.html">Initializer</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/fluid/evaluator.html">Evaluator</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/fluid/nets.html">Nets</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/fluid/optimizer.html">Optimizer</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/fluid/param_attr.html">ParamAttr</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/fluid/profiler.html">Profiler</a></li>
<li class="toctree-l3"><a class="reference internal" href="../api/v2/fluid/regularizer.html">Regularizer</a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../faq/index_cn.html">FAQ</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../faq/build_and_install/index_cn.html">编译安装与单元测试</a></li>
<li class="toctree-l2"><a class="reference internal" href="../faq/model/index_cn.html">模型配置</a></li>
<li class="toctree-l2"><a class="reference internal" href="../faq/parameter/index_cn.html">参数设置</a></li>
<li class="toctree-l2"><a class="reference internal" href="../faq/local/index_cn.html">本地训练与预测</a></li>
<li class="toctree-l2"><a class="reference internal" href="../faq/cluster/index_cn.html">集群训练与预测</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../mobile/index_cn.html">MOBILE</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../mobile/cross_compiling_for_android_cn.html">Android平台编译指南</a></li>
<li class="toctree-l2"><a class="reference internal" href="../mobile/cross_compiling_for_ios_cn.html">iOS平台编译指南</a></li>
<li class="toctree-l2"><a class="reference internal" href="../mobile/cross_compiling_for_raspberry_cn.html">Raspberry Pi平台编译指南</a></li>
</ul>
</li>
</ul>

        
    </nav>
    
    <section class="doc-content-wrap">

      
<div role="navigation" aria-label="breadcrumbs navigation">
  <ul class="wy-breadcrumbs">
      
    <li>Design Doc: Gradient Operators Registration</li>
  </ul>
</div>
      
      <div class="wy-nav-content" id="doc-content">
        <div class="rst-content">
          <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
           <div itemprop="articleBody">
            
  <div class="section" id="design-doc-gradient-operators-registration">
<span id="design-doc-gradient-operators-registration"></span><h1>Design Doc: Gradient Operators Registration<a class="headerlink" href="#design-doc-gradient-operators-registration" title="永久链接至标题">¶</a></h1>
<div class="section" id="the-problem-posed">
<span id="the-problem-posed"></span><h2>The Problem Posed<a class="headerlink" href="#the-problem-posed" title="永久链接至标题">¶</a></h2>
<p>Currently, for each C++ operator class definition, a <em>gradient operator creator</em> function is registered, which takes as input a C++ operator instance and returns the corresponding gradient operator instance.</p>
<p>However, we noticed two problems with the current design:</p>
<ol class="simple">
<li>As we decided to separate the <em>compilation</em> and the <em>execution</em> phases, we need to change the creator to take an <code class="docutils literal"><span class="pre">OpDesc</span></code> protobuf message in a <code class="docutils literal"><span class="pre">ProgramDesc</span></code> and inserts corresponding <code class="docutils literal"><span class="pre">OpDesc</span></code> messages into the <code class="docutils literal"><span class="pre">ProgramDesc</span></code> message.</li>
<li>For some operators, the gradient computation can be written in terms of existing operators.  For example, the gradient of <em>minus</em> operator consists of two operators &#8211; an <em>identity</em> operator followed by a <em>scale</em> operator.  Hence the registration mechanism needs to support mapping from an operator to a set of operators for the gradient computation.</li>
</ol>
</div>
<div class="section" id="the-current-implementation">
<span id="the-current-implementation"></span><h2>The Current Implementation<a class="headerlink" href="#the-current-implementation" title="永久链接至标题">¶</a></h2>
<p>Instances of the C++ class <code class="docutils literal"><span class="pre">OpInfo</span></code> are stored an associative map whose key is the operator type. The <code class="docutils literal"><span class="pre">grad_op_type</span></code> indicates the associated gradient operator type. An operator can create the gradient operator by invoking <code class="docutils literal"><span class="pre">OpInfo::creator_</span></code> of the gradient operator. The pseudo code is as follows</p>
<div class="highlight-cpp"><div class="highlight"><pre><span></span><span class="k">struct</span> <span class="n">OpInfo</span> <span class="p">{</span>
  <span class="n">std</span><span class="o">::</span><span class="n">function</span><span class="o">&lt;</span><span class="n">OperatorBase</span><span class="o">*</span><span class="p">(...)</span><span class="o">&gt;</span> <span class="n">creator_</span><span class="p">;</span>
  <span class="n">std</span><span class="o">::</span><span class="n">string</span> <span class="n">grad_op_type_</span><span class="p">;</span>
  <span class="p">...</span>
<span class="p">};</span>

<span class="n">map</span><span class="o">&lt;</span><span class="n">string</span><span class="p">,</span> <span class="n">OpInfo</span><span class="o">&gt;</span> <span class="n">OpInfoMap</span><span class="p">;</span>

<span class="n">OperatorBase</span><span class="o">*</span> <span class="nf">CreateGradientOperator</span><span class="p">(</span><span class="k">const</span> <span class="n">OperatorBase</span><span class="o">&amp;</span> <span class="n">op</span><span class="p">)</span> <span class="p">{</span>
  <span class="k">return</span> <span class="n">OpInfoMap</span><span class="p">.</span><span class="n">at</span><span class="p">(</span><span class="n">op</span><span class="p">.</span><span class="n">Type</span><span class="p">()).</span><span class="n">creator_</span><span class="p">(...);</span>
<span class="p">}</span>
</pre></div>
</div>
</div>
<div class="section" id="proposed-solution">
<span id="proposed-solution"></span><h2>Proposed Solution<a class="headerlink" href="#proposed-solution" title="永久链接至标题">¶</a></h2>
<p>The mapping relationship between an operator and its gradient operators is a function. The interface of this function is:</p>
<div class="highlight-cpp"><div class="highlight"><pre><span></span><span class="c1">// (OpDesc) --&gt; vector&lt;OpDesc&gt;</span>
<span class="n">std</span><span class="o">::</span><span class="n">function</span><span class="o">&lt;</span><span class="n">std</span><span class="o">::</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">OpDescBind</span><span class="o">&gt;</span><span class="p">(</span><span class="k">const</span> <span class="n">OpDescBind</span><span class="o">&amp;</span><span class="p">)</span><span class="o">&gt;</span><span class="p">;</span>
</pre></div>
</div>
<p>The function takes an <code class="docutils literal"><span class="pre">OpDescBind</span></code> of the forward operator and returns one or many gradient operator descriptions. <code class="docutils literal"><span class="pre">OpDescBind</span></code> is a C++ wrapper for  the protobuf message <code class="docutils literal"><span class="pre">OpDesc</span></code> for rapid manipulation of <code class="docutils literal"><span class="pre">OpDesc</span></code>.</p>
<p>The <code class="docutils literal"><span class="pre">GradOpDescMaker</span></code> will be registered in <code class="docutils literal"><span class="pre">OpInfo</span></code> and will replace the <code class="docutils literal"><span class="pre">grad_op_type_</span></code> field. The <code class="docutils literal"><span class="pre">OpInfo</span></code> should look like</p>
<div class="highlight-cpp"><div class="highlight"><pre><span></span><span class="k">struct</span> <span class="n">OpInfo</span> <span class="p">{</span>
  <span class="n">std</span><span class="o">::</span><span class="n">function</span><span class="o">&lt;</span><span class="n">std</span><span class="o">::</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">std</span><span class="o">::</span><span class="n">unique_ptr</span><span class="o">&lt;</span><span class="n">OpDescBind</span><span class="o">&gt;&gt;</span><span class="p">(</span><span class="k">const</span> <span class="n">OpDescBind</span><span class="o">&amp;</span><span class="p">)</span><span class="o">&gt;</span>  <span class="n">grad_op_maker_</span><span class="p">;</span>
  <span class="p">...</span>
<span class="p">};</span>
</pre></div>
</div>
<p>The <code class="docutils literal"><span class="pre">grad_op_maker_</span></code> is a <code class="docutils literal"><span class="pre">nullptr</span></code> if the operator does not have any associated gradient operators.</p>
<p>We propose a base class called <code class="docutils literal"><span class="pre">GradOpDescMakerBase</span></code> to let operator developers generate <code class="docutils literal"><span class="pre">Gradient</span> <span class="pre">Operators</span></code> easily. The public interface of that class is</p>
<div class="highlight-cpp"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">GradOpDescMakerBase</span> <span class="p">{</span>
<span class="k">public</span><span class="o">:</span>
  <span class="n">GradOpDescMakerBase</span><span class="p">(</span><span class="k">const</span> <span class="n">OpDescBind</span><span class="o">&amp;</span> <span class="p">);</span>
  <span class="k">virtual</span> <span class="n">std</span><span class="o">::</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">std</span><span class="o">::</span><span class="n">unique_ptr</span><span class="o">&lt;</span><span class="n">OpDescBind</span><span class="o">&gt;&gt;</span> <span class="k">operator</span><span class="p">()()</span><span class="k">const</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="p">};</span>
</pre></div>
</div>
<p>We can convert <code class="docutils literal"><span class="pre">GradOpDescMakerBase</span></code> to <code class="docutils literal"><span class="pre">std::function&lt;std::vector&lt;std::unique_ptr&lt;OpDescBind&gt;&gt;(const</span> <span class="pre">OpDescBind&amp;)&gt;</span></code> by</p>
<div class="highlight-cpp"><div class="highlight"><pre><span></span><span class="k">using</span> <span class="n">GradOpMaker</span> <span class="o">=</span> <span class="p">...;</span>
<span class="n">std</span><span class="o">::</span><span class="n">function</span><span class="o">&lt;</span><span class="n">std</span><span class="o">::</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">OpDescBind</span><span class="o">&gt;</span><span class="p">(</span><span class="k">const</span> <span class="n">OpDescBind</span><span class="o">&amp;</span><span class="p">)</span><span class="o">&gt;</span> <span class="n">func</span><span class="p">;</span>
<span class="n">func</span> <span class="o">=</span> <span class="p">[]</span> <span class="p">(</span><span class="k">const</span> <span class="n">OpDescBind</span><span class="o">&amp;</span> <span class="n">fwd_op</span><span class="p">)</span> <span class="p">{</span>
  <span class="n">GradOpMaker</span> <span class="n">maker</span><span class="p">(</span><span class="n">fwd_op</span><span class="p">);</span>
  <span class="k">return</span> <span class="nf">maker</span><span class="p">();</span>
<span class="p">};</span>
</pre></div>
</div>
<p>We can write many helper functions since the <code class="docutils literal"><span class="pre">GradOpDescMakerBase</span></code> is a class now. The basic helper functions get the variables of <code class="docutils literal"><span class="pre">Input</span></code>, <code class="docutils literal"><span class="pre">Output</span></code>, <code class="docutils literal"><span class="pre">InputGradient</span></code> and <code class="docutils literal"><span class="pre">OutputGradient</span></code> in the forwarding operator.</p>
<p>We should change register macros at the same time. In the current solution, there is no difference between forwarding operators and backward operators. So <code class="docutils literal"><span class="pre">REGISTER_OP</span></code> just register one operator. If the <code class="docutils literal"><span class="pre">REGISTER_OPERATOR</span></code> contains <code class="docutils literal"><span class="pre">OpProtoAndCheckerMaker</span></code> and <code class="docutils literal"><span class="pre">GradOpDescMaker</span></code>, we just list them in the same macro. It can be done by a macro contains <code class="docutils literal"><span class="pre">__VA_ARGS__</span></code>.</p>
<p>The user interface should be</p>
<div class="highlight-cpp"><div class="highlight"><pre><span></span><span class="n">vector</span><span class="o">&lt;</span><span class="n">OpDesc</span><span class="o">&gt;</span> <span class="n">MinusOpGradMaker</span><span class="p">(</span><span class="n">OpDesc</span><span class="p">)</span> <span class="p">{...}</span>
<span class="n">REGISTER_OPERATOR</span><span class="p">(</span><span class="n">minus</span><span class="p">,</span> <span class="n">MinusOp</span><span class="p">,</span> <span class="n">MinusOpProtoAndCheckerMaker</span><span class="p">,</span> <span class="n">SumOpGradMaker</span><span class="p">);</span>
<span class="c1">// Developers can still manually implement gradient operator.</span>
<span class="n">REGISTER_OPERATOR</span><span class="p">(</span><span class="n">minus_grad</span><span class="p">,</span> <span class="n">MinusGradOp</span><span class="p">);</span>
</pre></div>
</div>
<p>The interface of current <code class="docutils literal"><span class="pre">REGISTER_OP</span></code> macro could not be changed. In <code class="docutils literal"><span class="pre">REGISTER_OP</span></code>, it will invoke <code class="docutils literal"><span class="pre">REGISTER_OPERATOR</span></code> two times and generate GradOpDescMaker inside.</p>
<div class="highlight-cpp"><div class="highlight"><pre><span></span><span class="n">REGISTER_OP</span><span class="p">(</span><span class="n">minus</span><span class="p">,</span> <span class="n">MinusOp</span><span class="p">,</span> <span class="n">MinusOpProtoAndCheckerMaker</span><span class="p">,</span> <span class="n">minus_grad</span><span class="p">,</span> <span class="n">MinusGradOp</span><span class="p">);</span>
</pre></div>
</div>
</div>
</div>


           </div>
          </div>
          <footer>
  

  <hr/>

  <div role="contentinfo">
    <p>
        &copy; Copyright 2016, PaddlePaddle developers.

    </p>
  </div>
  Built with <a href="http://sphinx-doc.org/">Sphinx</a> using a <a href="https://github.com/snide/sphinx_rtd_theme">theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a>. 

</footer>

        </div>
      </div>

    </section>

  </div>
  

    <script type="text/javascript">
        var DOCUMENTATION_OPTIONS = {
            URL_ROOT:'../',
            VERSION:'',
            COLLAPSE_INDEX:false,
            FILE_SUFFIX:'.html',
            HAS_SOURCE:  true,
            SOURCELINK_SUFFIX: ".txt",
        };
    </script>
      <script type="text/javascript" src="../_static/jquery.js"></script>
      <script type="text/javascript" src="../_static/underscore.js"></script>
      <script type="text/javascript" src="../_static/doctools.js"></script>
      <script type="text/javascript" src="../_static/translations.js"></script>
      <script type="text/javascript" src="https://cdn.bootcss.com/mathjax/2.7.0/MathJax.js"></script>
       
  
    <script type="text/javascript" src="../_static/js/theme.js"></script>
  
  
  <script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/js/bootstrap.min.js" integrity="sha384-Tc5IQib027qvyjSMfHjOMaLkfuWVxZxUPnCJA7l2mCWNIpG9mGCD8wGNIcPD7Txa" crossorigin="anonymous"></script>
  <script src="https://cdn.jsdelivr.net/perfect-scrollbar/0.6.14/js/perfect-scrollbar.jquery.min.js"></script>
  <script src="../_static/js/paddle_doc_init.js"></script> 

</body>
</html>