program.html 23.4 KB
Newer Older
1 2 3 4 5 6 7 8 9 10


<!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">
  
11
  <title>Design Doc: PaddlePaddle Programs &mdash; PaddlePaddle  文档</title>
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124
  

  
  

  

  
  
    

  

  
  
    <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>
</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/docker_install_cn.html">PaddlePaddle的Docker容器使用方式</a></li>
<li class="toctree-l3"><a class="reference internal" href="../getstarted/build_and_install/cmake/build_from_source_cn.html">PaddlePaddle的编译选项</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>
125
<li class="toctree-l2"><a class="reference internal" href="../howto/usage/cluster/cluster_train_cn.html">PaddlePaddle分布式训练</a></li>
126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155
<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/build_cn.html">编译PaddlePaddle和运行单元测试</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></li>
<li class="toctree-l2"><a class="reference internal" href="../api/v2/run_logic.html">训练与应用</a></li>
</ul>
</li>
156 157 158 159 160 161 162 163
<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>
164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183
</ul>

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

      

 







<div role="navigation" aria-label="breadcrumbs navigation">
  <ul class="wy-breadcrumbs">
      
184
    <li>Design Doc: PaddlePaddle Programs</li>
185 186 187 188 189 190 191 192
  </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">
            
193 194 195 196 197 198
  <div class="section" id="design-doc-paddlepaddle-programs">
<span id="design-doc-paddlepaddle-programs"></span><h1>Design Doc: PaddlePaddle Programs<a class="headerlink" href="#design-doc-paddlepaddle-programs" title="永久链接至标题"></a></h1>
<div class="section" id="compile-and-execution">
<span id="compile-and-execution"></span><h2>Compile and Execution<a class="headerlink" href="#compile-and-execution" title="永久链接至标题"></a></h2>
<p>A PaddlePaddle program consists of two parts &#8211; the first generates a <code class="docutils literal"><span class="pre">ProgramDesc</span></code> protobuf message that describes the program, and the second runs this message using a C++ class <code class="docutils literal"><span class="pre">Executor</span></code>.</p>
<p>A simple example PaddlePaddle program can be found in <a class="reference internal" href="graph.html"><span class="doc">graph.md</span></a>:</p>
199 200 201 202 203 204 205 206
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="n">layer</span><span class="o">.</span><span class="n">data</span><span class="p">(</span><span class="s2">&quot;images&quot;</span><span class="p">)</span>
<span class="n">l</span> <span class="o">=</span> <span class="n">layer</span><span class="o">.</span><span class="n">data</span><span class="p">(</span><span class="s2">&quot;label&quot;</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">layer</span><span class="o">.</span><span class="n">fc</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">cost</span> <span class="o">=</span> <span class="n">layer</span><span class="o">.</span><span class="n">mse</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">l</span><span class="p">)</span>
<span class="n">optimize</span><span class="p">(</span><span class="n">cost</span><span class="p">)</span>
<span class="n">train</span><span class="p">(</span><span class="n">cost</span><span class="p">,</span> <span class="n">reader</span><span class="o">=</span><span class="n">mnist</span><span class="o">.</span><span class="n">train</span><span class="p">())</span>
</pre></div>
</div>
207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227
<p>The first five lines of the following PaddlePaddle program generates, or, compiles, the <code class="docutils literal"><span class="pre">ProgramDesc</span></code> message.  The last line runs it.</p>
</div>
<div class="section" id="programs-and-blocks">
<span id="programs-and-blocks"></span><h2>Programs and Blocks<a class="headerlink" href="#programs-and-blocks" title="永久链接至标题"></a></h2>
<p>The basic structure of a PaddlePaddle program is some nested blocks, as a C++ or Java program.</p>
<ul class="simple">
<li>program: some nested blocks</li>
<li><a class="reference internal" href="block.html"><span class="doc">block</span></a>:<ul>
<li>some local variable definitions, and</li>
<li>a sequence of operators</li>
</ul>
</li>
</ul>
<p>The concept of block comes from usual programs.  For example, the following C++ program has three blocks:</p>
<div class="highlight-c++"><div class="highlight"><pre><span></span><span class="kt">int</span> <span class="nf">main</span><span class="p">()</span> <span class="p">{</span> <span class="c1">// block 0</span>
  <span class="kt">int</span> <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span>
  <span class="k">if</span> <span class="p">(</span><span class="n">i</span> <span class="o">&lt;</span> <span class="mi">10</span><span class="p">)</span> <span class="p">{</span> <span class="c1">// block 1</span>
    <span class="k">for</span> <span class="p">(</span><span class="kt">int</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">10</span><span class="p">;</span> <span class="n">j</span><span class="o">++</span><span class="p">)</span> <span class="p">{</span> <span class="c1">// block 2</span>
    <span class="p">}</span>
  <span class="p">}</span>
  <span class="k">return</span> <span class="mi">0</span><span class="p">;</span>
228
<span class="p">}</span>
229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254
</pre></div>
</div>
<p>The following PaddlePaddle program has three blocks:</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">paddle</span> <span class="kn">as</span> <span class="nn">pd</span>  <span class="o">//</span> <span class="n">block</span> <span class="mi">0</span>

<span class="n">x</span> <span class="o">=</span> <span class="n">minibatch</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">30</span><span class="p">])</span> <span class="c1"># shape=[None, 1]</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">var</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># shape=[1], value=1</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">minibatch</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">30</span><span class="p">])</span> <span class="c1"># shape=[None, 1]</span>
<span class="n">cond</span> <span class="o">=</span> <span class="n">larger_than</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">15</span><span class="p">)</span> <span class="c1"># [false, true, true]</span>

<span class="n">ie</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">ifelse</span><span class="p">()</span>
<span class="k">with</span> <span class="n">ie</span><span class="o">.</span><span class="n">true_block</span><span class="p">():</span>  <span class="o">//</span> <span class="n">block</span> <span class="mi">1</span>
    <span class="n">d</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">layer</span><span class="o">.</span><span class="n">add_scalar</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
    <span class="n">ie</span><span class="o">.</span><span class="n">output</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">pd</span><span class="o">.</span><span class="n">layer</span><span class="o">.</span><span class="n">softmax</span><span class="p">(</span><span class="n">d</span><span class="p">))</span>
<span class="k">with</span> <span class="n">ie</span><span class="o">.</span><span class="n">false_block</span><span class="p">():</span>  <span class="o">//</span> <span class="n">block</span> <span class="mi">2</span>
    <span class="n">d</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">layer</span><span class="o">.</span><span class="n">fc</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
    <span class="n">ie</span><span class="o">.</span><span class="n">output</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">d</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="n">o1</span><span class="p">,</span> <span class="n">o2</span> <span class="o">=</span> <span class="n">ie</span><span class="p">(</span><span class="n">cond</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="blockdesc-and-programdesc">
<span id="blockdesc-and-programdesc"></span><h2><code class="docutils literal"><span class="pre">BlockDesc</span></code> and <code class="docutils literal"><span class="pre">ProgramDesc</span></code><a class="headerlink" href="#blockdesc-and-programdesc" title="永久链接至标题"></a></h2>
<p>All protobuf messages are defined in <code class="docutils literal"><span class="pre">framework.proto</span></code>.</p>
<p><code class="docutils literal"><span class="pre">BlockDesc</span></code> is straight-forward &#8211; it includes local variable definitions, <code class="docutils literal"><span class="pre">vars</span></code>, and a sequence of operators, <code class="docutils literal"><span class="pre">ops</span></code>.</p>
<div class="highlight-protobuf"><div class="highlight"><pre><span></span><span class="kd">message</span> <span class="nc">BlockDesc</span> <span class="p">{</span>
255 256 257 258
  <span class="k">required</span> <span class="kt">int32</span> <span class="na">parent</span> <span class="o">=</span> <span class="mi">1</span><span class="p">;</span>
  <span class="k">repeated</span> <span class="n">VarDesc</span> <span class="na">vars</span> <span class="o">=</span> <span class="mi">2</span><span class="p">;</span>
  <span class="k">repeated</span> <span class="n">OpDesc</span> <span class="na">ops</span> <span class="o">=</span> <span class="mi">3</span><span class="p">;</span>
<span class="p">}</span>
259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277
</pre></div>
</div>
<p>The parent ID indicates the parent block so that operators in a block can refer to variables defined locally and also those defined in their ancestor blocks.</p>
<p>All hierarchical blocks in a program are flattened and stored in an array. The block ID is the index of the block in this array.</p>
<div class="highlight-protobuf"><div class="highlight"><pre><span></span><span class="kd">message</span> <span class="nc">ProgramDesc</span> <span class="p">{</span>
  <span class="k">repeated</span> <span class="n">BlockDesc</span> <span class="na">blocks</span> <span class="o">=</span> <span class="mi">1</span><span class="p">;</span>
<span class="p">}</span>
</pre></div>
</div>
<div class="section" id="global-block">
<span id="global-block"></span><h3>Global Block<a class="headerlink" href="#global-block" title="永久链接至标题"></a></h3>
<p>The global block is the first one in the above array.</p>
</div>
</div>
<div class="section" id="operators-that-use-blocks">
<span id="operators-that-use-blocks"></span><h2>Operators that Use Blocks<a class="headerlink" href="#operators-that-use-blocks" title="永久链接至标题"></a></h2>
<p>In the above example, the operator <code class="docutils literal"><span class="pre">IfElseOp</span></code> has two blocks &#8211; the true branch and the false branch.</p>
<p>The definition of <code class="docutils literal"><span class="pre">OpDesc</span></code> shows that an operator could have some attributes:</p>
<div class="highlight-protobuf"><div class="highlight"><pre><span></span><span class="kd">message</span> <span class="nc">OpDesc</span> <span class="p">{</span>
278 279 280
  <span class="n">AttrDesc</span> <span class="na">attrs</span> <span class="o">=</span> <span class="mi">1</span><span class="p">;</span>
  <span class="o">...</span>
<span class="p">}</span>
281 282 283 284 285 286 287 288 289 290 291 292 293 294 295
</pre></div>
</div>
<p>and an attribute could be of type block, which is, in fact, a block ID as described above:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">message</span> <span class="n">AttrDesc</span> <span class="p">{</span>
  <span class="n">required</span> <span class="n">string</span> <span class="n">name</span> <span class="o">=</span> <span class="mi">1</span><span class="p">;</span>

  <span class="n">enum</span> <span class="n">AttrType</span> <span class="p">{</span>
    <span class="n">INT</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span>
    <span class="n">STRING</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span>
    <span class="o">...</span>
    <span class="n">BLOCK</span> <span class="o">=</span> <span class="o">...</span>
  <span class="p">}</span>
  <span class="n">required</span> <span class="n">AttrType</span> <span class="nb">type</span> <span class="o">=</span> <span class="mi">2</span><span class="p">;</span>

  <span class="n">optional</span> <span class="n">int32</span> <span class="n">block</span> <span class="o">=</span> <span class="mi">10</span><span class="p">;</span> <span class="o">//</span> <span class="n">when</span> <span class="nb">type</span> <span class="o">==</span> <span class="n">BLOCK</span>
296 297 298 299
  <span class="o">...</span>
<span class="p">}</span>
</pre></div>
</div>
300 301 302
</div>
<div class="section" id="infershape">
<span id="infershape"></span><h2>InferShape<a class="headerlink" href="#infershape" title="永久链接至标题"></a></h2>
303 304 305 306 307 308 309 310 311 312 313 314 315 316
<p>With this design, the InferShape function should take the following parameters:</p>
<div class="highlight-c++"><div class="highlight"><pre><span></span><span class="kt">void</span> <span class="nf">InferShape</span><span class="p">(</span><span class="kt">int</span> <span class="n">current_block</span><span class="p">,</span>
                <span class="kt">int</span> <span class="n">current_operator</span><span class="p">,</span>
                <span class="n">ProgramDesc</span><span class="o">*</span> <span class="n">program</span> <span class="c1">// might change VarDesc values.</span>
                <span class="p">)</span> <span class="p">{</span>
  <span class="p">...</span>
<span class="p">}</span>
</pre></div>
</div>
<p>where</p>
<ul class="simple">
<li><code class="docutils literal"><span class="pre">current_block</span></code> indices into <code class="docutils literal"><span class="pre">ProgramDesc::blocks</span></code>,</li>
<li><code class="docutils literal"><span class="pre">current_operator</span></code> indices into <code class="docutils literal"><span class="pre">BlockDesc::ops</span></code>.</li>
</ul>
317
</div>
318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377
</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>