parallel_do.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: Parallel_Do in PaddlePaddle &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"/> 
<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">

  <div class="wy-grid-for-nav">

    
    <nav data-toggle="wy-nav-shift" class="wy-nav-side">
      <div class="wy-side-scroll">
        <div class="wy-side-nav-search">
          

            <a href="../index_en.html" class="icon icon-home"> PaddlePaddle
          

          </a>

          
<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 class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="main navigation">
          
<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/quickstart_en.html">Quick Start</a></li>
<li class="toctree-l2"><a class="reference internal" href="../getstarted/concepts/use_concepts_en.html">Basic Concept</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../build_and_install/index_en.html">Install and Build</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../build_and_install/pip_install_en.html">Install using pip</a></li>
<li class="toctree-l2"><a class="reference internal" href="../build_and_install/docker_install_en.html">Run in Docker Containers</a></li>
<li class="toctree-l2"><a class="reference internal" href="../build_and_install/build_from_source_en.html">Build from Sources</a></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/cmd_parameter/index_en.html">Set Command-line Parameters</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../howto/cmd_parameter/use_case_en.html">Use Case</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/cmd_parameter/arguments_en.html">Argument Outline</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/cmd_parameter/detail_introduction_en.html">Detail Description</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="../howto/cluster/index_en.html">Distributed Training</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../howto/cluster/preparations_en.html">Preparations</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/cluster/cmd_argument_en.html">Command-line arguments</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/cluster/multi_cluster/index_en.html">Use different clusters</a><ul>
<li class="toctree-l4"><a class="reference internal" href="../howto/cluster/multi_cluster/fabric_en.html">Fabric</a></li>
<li class="toctree-l4"><a class="reference internal" href="../howto/cluster/multi_cluster/openmpi_en.html">OpenMPI</a></li>
<li class="toctree-l4"><a class="reference internal" href="../howto/cluster/multi_cluster/k8s_en.html">Kubernetes</a></li>
<li class="toctree-l4"><a class="reference internal" href="../howto/cluster/multi_cluster/k8s_aws_en.html">Kubernetes on AWS</a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="../howto/rnn/index_en.html">RNN Models</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../howto/rnn/rnn_config_en.html">RNN Configuration</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/rnn/recurrent_group_en.html">Recurrent Group Tutorial</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/rnn/hierarchical_layer_en.html">Layers supporting hierarchical sequence as input</a></li>
<li class="toctree-l3"><a class="reference internal" href="../howto/rnn/hrnn_rnn_api_compare_en.html">API comparision between RNN and hierarchical RNN</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="../dev/index_en.html">Development</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../dev/contribute_to_paddle_en.html">Contribute Code</a></li>
<li class="toctree-l2"><a class="reference internal" href="../dev/write_docs_en.html">Contribute Documentation</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../faq/index_en.html">FAQ</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../faq/build_and_install/index_en.html">Install, Build and Unit test</a></li>
<li class="toctree-l2"><a class="reference internal" href="../faq/model/index_en.html">Model Configuration</a></li>
<li class="toctree-l2"><a class="reference internal" href="../faq/parameter/index_en.html">Parameter Setting</a></li>
<li class="toctree-l2"><a class="reference internal" href="../faq/local/index_en.html">Local Training and Prediction</a></li>
<li class="toctree-l2"><a class="reference internal" href="../faq/cluster/index_en.html">Cluster Training and Prediction</a></li>
</ul>
</li>
</ul>

</nav>

        </div>
      </div>
    </nav>

    <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap">

      
      <nav class="wy-nav-top" role="navigation" aria-label="top navigation">
        <i data-toggle="wy-nav-top" class="fa fa-bars"></i>
        <a href="../index_en.html">PaddlePaddle</a>
      </nav>


      <div class="wy-nav-content">
        <div class="rst-content">
          

<div role="navigation" aria-label="breadcrumbs navigation">
  <ul class="wy-breadcrumbs">
    <li><a href="../index_en.html">Docs</a> &raquo;</li>
      
    <li>Design Doc: Parallel_Do in PaddlePaddle</li>
      <li class="wy-breadcrumbs-aside">
        
          
            <a href="../_sources/design/parallel_do.md.txt" rel="nofollow"> View page source</a>
          
        
      </li>
  </ul>
  <hr/>
</div>
          <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
           <div itemprop="articleBody">
            
  <div class="section" id="design-doc-parallel-do-in-paddlepaddle">
<span id="design-doc-parallel-do-in-paddlepaddle"></span><h1>Design Doc: Parallel_Do in PaddlePaddle<a class="headerlink" href="#design-doc-parallel-do-in-paddlepaddle" title="Permalink to this headline">¶</a></h1>
<p>In PaddlePaddle, we use parallel_do primitive to represent multithread data parallel processing.</p>
<div class="section" id="design-overview">
<span id="design-overview"></span><h2>Design overview<a class="headerlink" href="#design-overview" title="Permalink to this headline">¶</a></h2>
<p>The definition of a parallel_do op looks like the following</p>
<div class="highlight-c++"><div class="highlight"><pre><span></span><span class="n">AddInput</span><span class="p">(</span><span class="n">kInputs</span><span class="p">,</span> <span class="s">&quot;Inputs needed to be split onto different devices&quot;</span><span class="p">).</span><span class="n">AsDuplicable</span><span class="p">();</span>
<span class="n">AddInput</span><span class="p">(</span><span class="n">kParameters</span><span class="p">,</span> <span class="s">&quot;Parameters are duplicated over different devices&quot;</span><span class="p">)</span>
    <span class="p">.</span><span class="n">AsDuplicable</span><span class="p">();</span>
<span class="n">AddInput</span><span class="p">(</span><span class="n">kPlaces</span><span class="p">,</span> <span class="s">&quot;Devices used for parallel processing&quot;</span><span class="p">);</span>
<span class="n">AddOutput</span><span class="p">(</span><span class="n">kOutputs</span><span class="p">,</span> <span class="s">&quot;Outputs needed to be merged from different devices&quot;</span><span class="p">).</span><span class="n">AsDuplicable</span><span class="p">();</span>
<span class="n">AddOutput</span><span class="p">(</span><span class="n">kParallelScopes</span><span class="p">,</span>
          <span class="s">&quot;Scopes for all local variables in forward pass. One scope for each device&quot;</span><span class="p">);</span>
<span class="n">AddAttr</span><span class="o">&lt;</span><span class="n">framework</span><span class="o">::</span><span class="n">BlockDesc</span> <span class="o">*&gt;</span><span class="p">(</span><span class="n">kParallelBlock</span><span class="p">,</span>
                                <span class="s">&quot;List of operaters to be executed in parallel&quot;</span><span class="p">);</span>
</pre></div>
</div>
<p>A vanilla implementation of parallel_do can be shown as the following (<code class="docutils literal"><span class="pre">|</span></code> means single thread and
<code class="docutils literal"><span class="pre">||||</span></code> means multiple threads)</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">In</span> <span class="n">the</span> <span class="n">forward</span> <span class="k">pass</span>
  <span class="o">|</span>      <span class="n">Split</span> <span class="nb">input</span> <span class="n">onto</span> <span class="n">different</span> <span class="n">devices</span>
  <span class="o">|</span>      <span class="n">Copy</span> <span class="n">parameter</span> <span class="n">onto</span> <span class="n">different</span> <span class="n">devices</span>
  <span class="o">||||</span>   <span class="n">Compute</span> <span class="n">forward</span> <span class="k">pass</span> <span class="ow">in</span> <span class="n">parallel</span>
  <span class="o">|</span>      <span class="n">Merge</span> <span class="n">output</span> <span class="kn">from</span> <span class="nn">different</span> <span class="n">devices</span>

<span class="n">In</span> <span class="n">the</span> <span class="n">backward</span> <span class="k">pass</span>
  <span class="o">|</span>      <span class="n">Split</span> <span class="n">output</span><span class="nd">@grad</span> <span class="n">onto</span> <span class="n">different</span> <span class="n">devices</span>
  <span class="o">||||</span>   <span class="n">Compute</span> <span class="n">backward</span> <span class="k">pass</span> <span class="ow">in</span> <span class="n">parallel</span>
  <span class="o">|</span>      <span class="n">accumulate</span> <span class="n">param</span><span class="nd">@grad</span> <span class="kn">from</span> <span class="nn">different</span> <span class="n">devices</span> <span class="n">to</span> <span class="n">the</span> <span class="n">first</span> <span class="n">device</span>
  <span class="o">|</span>      <span class="n">Merge</span> <span class="nb">input</span><span class="nd">@grad</span> <span class="kn">from</span> <span class="nn">different</span> <span class="n">devices</span>
  <span class="o">|</span>      <span class="n">Copy</span> <span class="n">param</span><span class="nd">@grad</span> <span class="n">to</span> <span class="n">the</span> <span class="n">place</span> <span class="n">of</span> <span class="n">parallel_do_op</span>
</pre></div>
</div>
<p>This implementation allows to write mixed device program like this</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">W1</span> <span class="o">=</span> <span class="n">fluid</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="p">[</span><span class="mi">100</span><span class="p">,</span><span class="mi">20</span><span class="p">],</span> <span class="n">parameter</span><span class="o">=</span><span class="n">true</span><span class="p">)</span>
<span class="n">W2</span> <span class="o">=</span> <span class="n">fluid</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="p">[</span><span class="mi">20</span><span class="p">,</span><span class="mi">15</span><span class="p">],</span> <span class="n">parameter</span><span class="o">=</span><span class="n">true</span><span class="p">)</span>

<span class="n">data</span> <span class="o">=</span> <span class="n">layers</span><span class="o">.</span><span class="n">data</span><span class="p">()</span>

<span class="n">gpu_places</span> <span class="o">=</span> <span class="n">layers</span><span class="o">.</span><span class="n">get_place</span><span class="p">(</span><span class="n">use_gpu</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="c1"># parallel processing on multiple GPUs</span>
<span class="n">pd</span> <span class="o">=</span> <span class="n">ParallelDo</span><span class="p">(</span><span class="n">gpu_places</span><span class="p">)</span>
<span class="k">with</span> <span class="n">pd</span><span class="o">.</span><span class="n">do</span><span class="p">(</span><span class="nb">input</span><span class="o">=</span><span class="n">data</span><span class="p">):</span>
    <span class="n">prediction</span> <span class="o">=</span> <span class="n">softmax</span><span class="p">(</span><span class="n">fc</span><span class="p">(</span><span class="n">fc</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">W1</span><span class="p">),</span> <span class="n">W2</span><span class="p">))</span>
    <span class="n">write_output</span><span class="p">(</span><span class="n">prediction</span><span class="p">)</span>
<span class="n">prediction</span> <span class="o">=</span> <span class="n">pd</span><span class="p">()</span>
<span class="n">loss</span> <span class="o">=</span> <span class="n">cross_entropy</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">label</span><span class="p">)</span>
</pre></div>
</div>
<p>And the programDesc are like the following</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="c1"># start_program will be run by executor(CPUPlace), all w1, w2 will be allocated on CPU</span>
<span class="n">start_program</span>
<span class="p">{</span>
  <span class="nb">vars</span><span class="p">:</span> <span class="n">w1</span><span class="p">,</span> <span class="n">w2</span>
  <span class="n">ops</span><span class="p">:</span> <span class="n">init</span><span class="p">(</span><span class="n">w1</span><span class="p">),</span> <span class="n">init</span><span class="p">(</span><span class="n">w2</span><span class="p">)</span>
<span class="p">}</span>

<span class="n">main_program</span>
<span class="p">{</span>
<span class="n">block0</span> <span class="p">{</span>
  <span class="nb">vars</span><span class="p">:</span> <span class="n">data</span><span class="p">,</span> <span class="n">places</span><span class="p">,</span> <span class="n">w1</span><span class="p">,</span> <span class="n">w2</span><span class="p">,</span> <span class="n">w1_grad</span><span class="p">,</span> <span class="n">w2_grad</span><span class="p">,</span>
  <span class="n">ops</span><span class="p">:</span> <span class="n">data</span><span class="p">,</span> <span class="n">get_place</span><span class="p">,</span> <span class="n">parallel_do</span><span class="p">(</span><span class="n">block1</span><span class="p">),</span>
       <span class="n">parallel_do_grad</span><span class="p">(</span><span class="n">block2</span><span class="p">),</span>
       <span class="n">sgd</span><span class="p">(</span><span class="n">w2</span><span class="p">,</span> <span class="n">w2_grad</span><span class="p">),</span>
       <span class="n">sgd</span><span class="p">(</span><span class="n">w1</span><span class="p">,</span> <span class="n">w1_grad</span><span class="p">)</span>
<span class="p">}</span>
<span class="n">block1</span> <span class="p">{</span> <span class="c1"># the forward pass</span>
  <span class="n">parent_block</span><span class="p">:</span> <span class="mi">0</span>
  <span class="nb">vars</span><span class="p">:</span> <span class="n">data</span><span class="p">,</span> <span class="n">h1</span><span class="p">,</span> <span class="n">h2</span><span class="p">,</span> <span class="n">loss</span>
  <span class="n">ops</span><span class="p">:</span> <span class="n">fc</span><span class="p">,</span> <span class="n">fc</span><span class="p">,</span> <span class="n">softmax</span>
<span class="p">}</span>
<span class="n">block2</span> <span class="p">{</span> <span class="c1"># the backward pass</span>
  <span class="n">parent_block</span><span class="p">:</span> <span class="mi">1</span>
  <span class="nb">vars</span><span class="p">:</span> <span class="n">data_grad</span><span class="p">,</span> <span class="n">h1_grad</span><span class="p">,</span> <span class="n">h2_grad</span><span class="p">,</span> <span class="n">loss_gard</span><span class="p">,</span> <span class="n">local_w1_grad</span><span class="p">,</span> <span class="n">local_w2_grad</span>
  <span class="n">ops</span><span class="p">:</span> <span class="n">softmax_grad</span><span class="p">,</span>
       <span class="n">fc_grad</span>
       <span class="n">fc_grad</span>
<span class="p">}</span>
<span class="p">}</span>
</pre></div>
</div>
</div>
<div class="section" id="performance-imporvement">
<span id="performance-imporvement"></span><h2>Performance Imporvement<a class="headerlink" href="#performance-imporvement" title="Permalink to this headline">¶</a></h2>
<p>There are serial places we can make this parallel_do faster.</p>
<div class="section" id="forward-split-input-onto-different-devices">
<span id="forward-split-input-onto-different-devices"></span><h3>forward: split input onto different devices<a class="headerlink" href="#forward-split-input-onto-different-devices" title="Permalink to this headline">¶</a></h3>
<p>If the input of the parallel_do is independent from any prior opeartors, we can avoid this step by
prefetching the input onto different devices in a seperate background thread. And the python code
looks like this.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span>pd = ParallelDo(gpu_places)
with pd.do():
    feature = get_data_from_prefetch_queue(gpu_places)
    prediction = my_net(feature)
    write_output(activation)
</pre></div>
</div>
</div>
<div class="section" id="forward-copy-parameter-to-onto-different-devices">
<span id="forward-copy-parameter-to-onto-different-devices"></span><h3>forward: Copy parameter to onto different devices<a class="headerlink" href="#forward-copy-parameter-to-onto-different-devices" title="Permalink to this headline">¶</a></h3>
<p>We can avoid this step by making each device have a copy of the parameter. This requires:</p>
<ol class="simple">
<li><code class="docutils literal"><span class="pre">fluid.default_start_up_program()</span></code> to be run on all devices</li>
<li>In the backward, allreduce param&#64;grad at different devices, this requires<ol>
<li><code class="docutils literal"><span class="pre">backward.py</span></code> add <code class="docutils literal"><span class="pre">allreduce</span></code> operators at parallel_do_grad</li>
<li><code class="docutils literal"><span class="pre">allreduce</span></code> operators need to be called in async mode to achieve maximum throughput</li>
</ol>
</li>
<li>apply gradients related op(i.e. cliping, normalization, decay, sgd) on different devices in parallel</li>
</ol>
<p>By doing so, we also avoided &#8220;backward: accumulate param&#64;grad from different devices to the first device&#8221;.
And the ProgramDesc looks like the following</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="c1"># w1, w2 will be allocated on all GPUs</span>
<span class="n">start_program</span>
<span class="p">{</span>
<span class="n">block0</span> <span class="p">{</span>
  <span class="n">parallel_do</span><span class="p">(</span><span class="n">block1</span><span class="p">)</span>
<span class="p">}</span>
<span class="n">block1</span> <span class="p">{</span>
  <span class="n">parent_block</span><span class="p">:</span> <span class="mi">0</span>
  <span class="nb">vars</span><span class="p">:</span> <span class="n">w1</span><span class="p">,</span> <span class="n">w2</span>
  <span class="n">ops</span><span class="p">:</span> <span class="n">init</span><span class="p">(</span><span class="n">w1</span><span class="p">),</span> <span class="n">init</span><span class="p">(</span><span class="n">w2</span><span class="p">)</span>
<span class="p">}</span>
<span class="p">}</span>

<span class="n">main_program</span>
<span class="p">{</span>
<span class="n">block0</span> <span class="p">{</span>
  <span class="nb">vars</span><span class="p">:</span> <span class="n">data</span><span class="p">,</span> <span class="n">places</span><span class="p">,</span> <span class="n">w1</span><span class="p">,</span> <span class="n">w2</span>
  <span class="n">ops</span><span class="p">:</span> <span class="n">data</span><span class="p">,</span> <span class="n">get_place</span><span class="p">,</span> <span class="n">parallel_do</span><span class="p">(</span><span class="n">block1</span><span class="p">),</span>
       <span class="n">parallel_do_grad</span><span class="p">(</span><span class="n">block2</span><span class="p">),</span>      <span class="c1"># append_backward</span>
       <span class="n">parallel_do</span><span class="p">(</span><span class="n">block3</span><span class="p">)</span>            <span class="c1"># append_optimization</span>
       
<span class="p">}</span>
<span class="n">block1</span> <span class="p">{</span>
  <span class="n">parent_block</span><span class="p">:</span> <span class="mi">0</span>
  <span class="nb">vars</span><span class="p">:</span> <span class="n">data</span><span class="p">,</span> <span class="n">h1</span><span class="p">,</span> <span class="n">h2</span><span class="p">,</span> <span class="n">loss</span>
  <span class="n">ops</span><span class="p">:</span> <span class="n">fc</span><span class="p">,</span> <span class="n">fc</span><span class="p">,</span> <span class="n">softmax</span>
<span class="p">}</span>
<span class="n">block2</span> <span class="p">{</span>
  <span class="n">parent_block</span><span class="p">:</span> <span class="mi">1</span>
  <span class="nb">vars</span><span class="p">:</span> <span class="n">data_grad</span><span class="p">,</span> <span class="n">h1_grad</span><span class="p">,</span> <span class="n">h2_grad</span><span class="p">,</span> <span class="n">loss_gard</span><span class="p">,</span> <span class="n">w1_grad</span><span class="p">,</span> <span class="n">w2_grad</span>
  <span class="n">ops</span><span class="p">:</span> <span class="n">softmax_grad</span><span class="p">,</span>
       <span class="n">fc_grad</span><span class="p">,</span> <span class="n">allreduce</span><span class="p">(</span><span class="n">places</span><span class="p">,</span> <span class="n">scopes</span><span class="p">,</span> <span class="n">w1_grad</span><span class="p">),</span>
       <span class="n">fc_grad</span><span class="p">,</span> <span class="n">allreduce</span><span class="p">(</span><span class="n">places</span><span class="p">,</span> <span class="n">scopes</span><span class="p">,</span> <span class="n">w2_grad</span><span class="p">)</span>
<span class="p">}</span>
<span class="n">block3</span> <span class="p">{</span>
  <span class="n">parent_block</span><span class="p">:</span> <span class="mi">0</span>
  <span class="nb">vars</span><span class="p">:</span> <span class="n">lr</span>
  <span class="n">ops</span><span class="p">:</span> <span class="n">sgd</span><span class="p">(</span><span class="n">w2</span><span class="p">,</span> <span class="n">w2_grad</span><span class="p">),</span>
       <span class="n">sgd</span><span class="p">(</span><span class="n">w1</span><span class="p">,</span> <span class="n">w1_grad</span><span class="p">)</span>
<span class="p">}</span>
<span class="p">}</span>
</pre></div>
</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
        };
    </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 type="text/javascript">
      jQuery(function () {
          SphinxRtdTheme.StickyNav.enable();
      });
  </script>
   

</body>
</html>