Skip to content

P
Paddle

PaddlePaddle / Paddle
大约 1 年 前同步成功

通知 2298
Star 20931
Fork 5422
P
Paddle
切换分支/标签
查找文件 普通视图历史永久链接Permalink
functions_operators_layers.html 23.2 KB
Newer Older
T
Deploy to GitHub Pages: 1ff47b284c565d030b198705d5f18b4bd4ce53e5  
Travis CI 已提交
1 2 3 4 5 6 7 8 9 10 11 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 


<!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: Functions, Operators, and Layers &mdash; PaddlePaddle  documentation</title>
  

  
  

  

  
  
    

  

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

  
  
        <link rel="index" title="Index"
              href="../genindex.html"/>
        <link rel="search" title="Search" href="../search.html"/>
    <link rel="top" title="PaddlePaddle  documentation" 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_en.html">GET STARTED</a></li>
<li class="toctree-l1"><a class="reference internal" href="../howto/index_en.html">HOW TO</a></li>
<li class="toctree-l1"><a class="reference internal" href="../api/index_en.html">API</a></li>
<li class="toctree-l1"><a class="reference internal" href="../mobile/index_en.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_en.html">GET STARTED</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../getstarted/build_and_install/index_en.html">Install and Build</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../getstarted/build_and_install/pip_install_en.html">Install Using pip</a></li>
<li class="toctree-l3"><a class="reference internal" href="../getstarted/build_and_install/docker_install_en.html">Run in Docker Containers</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/dev/build_en.html">Build using Docker</a></li>
<li class="toctree-l3"><a class="reference internal" href="../getstarted/build_and_install/build_from_source_en.html">Build from Sources</a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../howto/index_en.html">HOW TO</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../howto/usage/cmd_parameter/index_en.html">Set Command-line Parameters</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../howto/usage/cmd_parameter/use_case_en.html">Use Case</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/usage/cmd_parameter/arguments_en.html">Argument Outline</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/usage/cmd_parameter/detail_introduction_en.html">Detail Description</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="../howto/usage/cluster/cluster_train_en.html">PaddlePaddle Distributed Training</a></li>
<li class="toctree-l2"><a class="reference internal" href="../howto/usage/k8s/k8s_en.html">Paddle On Kubernetes</a></li>
<li class="toctree-l2"><a class="reference internal" href="../howto/usage/k8s/k8s_aws_en.html">Distributed PaddlePaddle Training on AWS with Kubernetes</a></li>
<li class="toctree-l2"><a class="reference internal" href="../howto/dev/new_layer_en.html">Write New Layers</a></li>
<li class="toctree-l2"><a class="reference internal" href="../howto/dev/contribute_to_paddle_en.html">Contribute Code</a></li>
<li class="toctree-l2"><a class="reference internal" href="../howto/dev/write_docs_en.html">Contribute Documentation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../howto/deep_model/rnn/index_en.html">RNN Models</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../howto/deep_model/rnn/rnn_config_en.html">RNN Configuration</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="../howto/optimization/gpu_profiling_en.html">Tune GPU Performance</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../api/index_en.html">API</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../api/v2/model_configs.html">Model Configuration</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">Data Reader Interface and DataSets</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">Training and Inference</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="../mobile/index_en.html">MOBILE</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../mobile/cross_compiling_for_android_en.html">Build PaddlePaddle for Android</a></li>
<li class="toctree-l2"><a class="reference internal" href="../mobile/cross_compiling_for_raspberry_en.html">Build PaddlePaddle for Raspberry Pi</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: Functions, Operators, and Layers</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-functions-operators-and-layers">
<span id="design-doc-functions-operators-and-layers"></span><h1>Design Doc: Functions, Operators, and Layers<a class="headerlink" href="#design-doc-functions-operators-and-layers" title="Permalink to this headline"></a></h1>
<p>In a DL system, we can compose one or more fine grained operators into a coarse grained one.  For example, the FC layer can be composed of a multiplication operator and an add operator.</p>
<p>Historically, some fine grained operations are known as operators, and some coarse level ones are known as layers.  But we need a well-defined separation.</p>
<p>In general, operators are those very fine grained operations, e.g., mul and add. In the implementation, we can write them as C++ functions:</p>
<div class="highlight-c++"><div class="highlight"><pre><span></span><span class="k">template</span> <span class="o">&lt;</span><span class="k">typename</span> <span class="n">T</span><span class="o">&gt;</span> <span class="n">T</span> <span class="n">add</span><span class="p">(</span><span class="n">T</span> <span class="n">x</span><span class="p">,</span> <span class="n">T</span> <span class="n">y</span><span class="p">)</span> <span class="p">{</span> <span class="k">return</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">;</span> <span class="p">}</span>
<span class="k">template</span> <span class="o">&lt;</span><span class="k">typename</span> <span class="n">T</span><span class="o">&gt;</span> <span class="n">T</span> <span class="n">mul</span><span class="p">(</span><span class="n">T</span> <span class="n">x</span><span class="p">,</span> <span class="n">T</span> <span class="n">y</span><span class="p">)</span> <span class="p">{</span> <span class="k">return</span> <span class="n">x</span> <span class="o">*</span> <span class="n">y</span><span class="p">;</span> <span class="p">}</span>
</pre></div>
</div>
<p>Then we can wrap them into operators which are C++ classes and can be created from Python bindings by name.  A C macro can do this. For example, the following macro invocation</p>
<div class="highlight-c++"><div class="highlight"><pre><span></span><span class="cp">#define MAKE_FUNCTION_OPERATOR(mul);</span>
</pre></div>
</div>
<p>generates</p>
<div class="highlight-c++"><div class="highlight"><pre><span></span><span class="k">template</span> <span class="o">&lt;</span><span class="k">typename</span> <span class="n">T</span><span class="o">&gt;</span> <span class="k">class</span> <span class="nc">mulOp</span> <span class="o">:</span> <span class="k">public</span> <span class="n">OperatorBase</span> <span class="p">{...};</span>
<span class="n">REGISTER_OP</span><span class="p">(</span><span class="n">mulOp</span><span class="o">&lt;</span><span class="n">float32</span><span class="o">&gt;</span><span class="p">,</span> <span class="s">&quot;mul&quot;</span><span class="p">);</span>
</pre></div>
</div>
<p>so that in Python we can create operator mul by:</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">X1</span> <span class="o">=</span> <span class="n">Var</span><span class="p">()</span>
<span class="n">X2</span> <span class="o">=</span> <span class="n">Var</span><span class="p">()</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">Var</span><span class="p">()</span>
<span class="n">paddle</span><span class="o">.</span><span class="n">cpp</span><span class="o">.</span><span class="n">create_operator</span><span class="p">(</span><span class="s2">&quot;mul&quot;</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="p">[</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">],</span> <span class="n">output</span><span class="o">=</span><span class="n">Y</span><span class="p">)</span>
</pre></div>
</div>
<p>Also, at the same time, we can compose a coarse level C++ operator class by composing functions <code class="docutils literal"><span class="pre">mul</span></code> and <code class="docutils literal"><span class="pre">add</span></code>:</p>
<div class="highlight-c++"><div class="highlight"><pre><span></span><span class="k">template</span> <span class="o">&lt;</span><span class="k">typename</span> <span class="n">T</span><span class="o">&gt;</span>
<span class="k">class</span> <span class="nc">FCOp</span> <span class="o">:</span> <span class="k">public</span> <span class="n">OperatorBase</span> <span class="p">{</span>
 <span class="k">public</span><span class="o">:</span>
  <span class="kt">void</span> <span class="n">Run</span><span class="p">(...)</span> <span class="p">{</span>
    <span class="n">add</span><span class="p">(</span><span class="n">mul</span><span class="p">(</span><span class="n">Input</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="p">(</span><span class="s">&quot;X&quot;</span><span class="p">),</span> <span class="n">Input</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="p">(</span><span class="s">&quot;W&quot;</span><span class="p">)),</span> <span class="n">Input</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="p">(</span><span class="s">&quot;b&quot;</span><span class="p">);</span>
  <span class="p">}</span>
<span class="p">};</span>
<span class="n">REGISTER_OP</span><span class="p">(</span><span class="n">FCOp</span><span class="p">,</span> <span class="s">&quot;fc&quot;</span><span class="p">);</span>
</pre></div>
</div>
<p>We need to support such composition in Python as well.  To do so, we need a higher level Python wrapping of operator creation than <code class="docutils literal"><span class="pre">paddle.cpp.create_operator</span></code>.  This higher level operator API should be compatible with the layer API.</p>
<p>Let&#8217;s explain using an example.  Suppose that we are going to compose the FC using mul and add in Python, we&#8217;d like to have Python functions <code class="docutils literal"><span class="pre">mul</span></code> and <code class="docutils literal"><span class="pre">add</span></code> defined in module <code class="docutils literal"><span class="pre">operator</span></code>:</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">operator</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">):</span>
    <span class="n">O</span> <span class="o">=</span> <span class="n">Var</span><span class="p">()</span>
    <span class="n">paddle</span><span class="o">.</span><span class="n">cpp</span><span class="o">.</span><span class="n">create_operator</span><span class="p">(</span><span class="s2">&quot;mul&quot;</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="p">{</span><span class="n">X1</span><span class="p">,</span> <span class="n">Y1</span><span class="p">},</span> <span class="n">output</span><span class="o">=</span><span class="n">O</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">O</span>

<span class="k">def</span> <span class="nf">operator</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">):</span>
    <span class="n">O</span> <span class="o">=</span> <span class="n">Var</span><span class="p">()</span>
    <span class="n">paddle</span><span class="o">.</span><span class="n">cpp</span><span class="o">.</span><span class="n">create_operator</span><span class="p">(</span><span class="s2">&quot;add&quot;</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="p">{</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">},</span> <span class="n">output</span><span class="o">=</span><span class="n">O</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">O</span>
</pre></div>
</div>
<p>Above code snippets are automatically generated.  Given them, users can define</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">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">W</span> <span class="o">=</span> <span class="n">Var</span><span class="p">()</span>
    <span class="n">b</span> <span class="o">=</span> <span class="n">Var</span><span class="p">()</span>
    <span class="k">return</span> <span class="n">operator</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">operator</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">W</span><span class="p">),</span> <span class="n">b</span><span class="p">)</span>
</pre></div>
</div>
<p>If we don&#8217;t have <code class="docutils literal"><span class="pre">operator.mul</span></code> and <code class="docutils literal"><span class="pre">operator.add</span></code>, the definiton of <code class="docutils literal"><span class="pre">layer.fc</span></code> would be complicated:</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">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">W</span> <span class="o">=</span> <span class="n">Var</span><span class="p">()</span>
    <span class="n">b</span> <span class="o">=</span> <span class="n">Var</span><span class="p">()</span>
    <span class="n">O1</span> <span class="o">=</span> <span class="n">Var</span><span class="p">()</span>
    <span class="n">paddle</span><span class="o">.</span><span class="n">cpp</span><span class="o">.</span><span class="n">create_operator</span><span class="p">(</span><span class="s2">&quot;mul&quot;</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="p">[</span><span class="n">X</span><span class="p">,</span> <span class="n">W</span><span class="p">],</span> <span class="n">output</span><span class="o">=</span><span class="n">O1</span><span class="p">)</span>
    <span class="n">O2</span> <span class="o">=</span> <span class="n">Var</span><span class="p">()</span>
    <span class="n">paddle</span><span class="o">.</span><span class="n">cpp</span><span class="o">.</span><span class="n">create_operator</span><span class="p">(</span><span class="s2">&quot;add&quot;</span><span class="p">,</span> <span class="nb">input</span><span class="o">=</span><span class="p">[</span><span class="n">O1</span><span class="p">,</span> <span class="n">b</span><span class="p">],</span> <span class="n">output</span><span class="o">=</span><span class="n">O2</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">O2</span>
</pre></div>
</div>
<p>We&#8217;d like to have Python bindings to operators in package <code class="docutils literal"><span class="pre">paddle.operator</span></code>, and Python compositions of operators in package <code class="docutils literal"><span class="pre">paddle.layer</span></code>.  So we have the following concepts in above illustrative example:</p>
<p>| C++ functions/functors | mul          | add          |             |          |
|&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;|&#8212;&#8212;&#8212;&#8212;&#8211;|&#8212;&#8212;&#8212;&#8212;&#8211;|&#8212;&#8212;&#8212;&#8212;-|&#8212;&#8212;&#8212;-|
| C++ operator class     | mulOp        | addOp        | FCOp        |          |
| Python binding         | operator.mul | operator.add | operator.fc |          |
| Python function        |              |              |             | layer.fc |</p>
<p>This is how we differentiate layer and operators in PaddlePaddle:</p>
<ul class="simple">
<li>those defined in C++ and have a lightweighted Python wrapper in module <code class="docutils literal"><span class="pre">operators</span></code> are operators; whereas</li>
<li>those who don&#8217;t have C++ implementations but a Python implementation that compose C++ operators are known as layers.</li>
</ul>
</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="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></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>