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<p>There are two typical ways to implement autograd, via symbolic differentiation like <a href="http://deeplearning.net/software/theano/index.html">Theano</a> or reverse differentiation like <a href="https://pytorch.org/docs/stable/notes/autograd.html">Pytorch</a>. Singa follows Pytorch way, which records the computation graph and apply the backward propagation automatically after forward propagation. The autograd algorithm is explained in details <a href="https://pytorch.org/docs/stable/notes/autograd.html">here</a>. We explain the relevant modules in Singa and give an example to illustrate the usage.</p>
<h2><a class="anchor" aria-hidden="true" id="relevant-modules"></a><a href="#relevant-modules" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Relevant Modules</h2>
<p>There are three classes involved in autograd, namely <code>singa.tensor.Tensor</code>, <code>singa.autograd.Operation</code>, and <code>singa.autograd.Layer</code>. In the rest of this article, we use tensor, operation and layer to refer to an instance of the respective class.</p>
<h3><a class="anchor" aria-hidden="true" id="tensor"></a><a href="#tensor" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Tensor</h3>
<p>Three attributes of Tensor are used by autograd,</p>
<ul>
<li><code>.creator</code> is an <code>Operation</code> instance. It records the operation that generates the Tensor instance.</li>
<li><code>.requires_grad</code> is a boolean variable. It is used to indicate that the autograd algorithm needs to compute the gradient of the tensor (i.e., the owner). For example, during backpropagation, the gradients of the tensors for the weight matrix of a linear layer and the feature maps of a convolution layer (not the bottom layer) should be computed.</li>
<li><code>.stores_grad</code> is a boolean variable. It is used to indicate that the gradient of the owner tensor should be stored and output by the backward function. For example, the gradient of the feature maps is computed during backpropagation, but is not included in the output of the backward function.</li>
</ul>
<p>Programmers can change <code>requires_grad</code> and <code>stores_grad</code> of a Tensor instance. For example, if later is set to True, the corresponding gradient is included in the output of the backward function. It should be noted that if <code>stores_grad</code> is True, then <code>requires_grad</code> must be true, not vice versa.</p>
<h3><a class="anchor" aria-hidden="true" id="operation"></a><a href="#operation" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Operation</h3>
<p>It takes one or more <code>Tensor</code> instances as input, and then outputs one or more <code>Tensor</code> instances. For example, ReLU can be implemented as a specific Operation subclass. When an <code>Operation</code> instance is called (after instantiation), the following two steps are executed:</p>
<ol>
<li>record the source operations, i.e., the <code>creator</code>s of the input tensors.</li>
<li>do calculation by calling member function <code>.forward()</code></li>
</ol>
<p>There are two member functions for forwarding and backwarding, i.e., <code>.forward()</code> and <code>.backward()</code>. They take <code>Tensor.data</code> as inputs (the type is <code>CTensor</code>), and output <code>Ctensor</code>s. To add a specific operation, subclass <code>operation</code> should implement their own <code>.forward()</code> and <code>.backward()</code>. The <code>backward()</code> function is called by the <code>backward()</code> function of autograd automatically during backward propogation to compute the gradients of inputs (according to the <code>require_grad</code> field).</p>
<h3><a class="anchor" aria-hidden="true" id="layer"></a><a href="#layer" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Layer</h3>
<p>For those operations that require parameters, we package them into a new class, <code>Layer</code>. For example, convolution operation is wrapped into a convolution layer. <code>Layer</code> manages (stores) the parameters and calls the corresponding <code>Operation</code>s to implement the transformation.</p>
<h2><a class="anchor" aria-hidden="true" id="examples"></a><a href="#examples" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Examples</h2>
<p>Multiple examples are provided in the <a href="https://github.com/apache/singa/tree/master/examples/autograd">example folder</a>. We explain two representative examples here.</p>
<h3><a class="anchor" aria-hidden="true" id="operation-only"></a><a href="#operation-only" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Operation only</h3>
<p>The following codes implement a MLP model using only Operation instances (no Layer instances).</p>
<h4><a class="anchor" aria-hidden="true" id="import-packages"></a><a href="#import-packages" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Import packages</h4>
<pre><code class="hljs css language-python"><span class="hljs-keyword">from</span> singa.tensor <span class="hljs-keyword">import</span> Tensor
<span class="hljs-keyword">from</span> singa <span class="hljs-keyword">import</span> autograd
<span class="hljs-keyword">from</span> singa <span class="hljs-keyword">import</span> opt
</code></pre>
<h4><a class="anchor" aria-hidden="true" id="create-weight-matrix-and-bias-vector"></a><a href="#create-weight-matrix-and-bias-vector" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Create weight matrix and bias vector</h4>
<p>The parameter tensors are created with both <code>requires_grad</code> and <code>stores_grad</code> set to <code>True</code>.</p>
<pre><code class="hljs css language-python">w0 = Tensor(shape=(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>), requires_grad=<span class="hljs-literal">True</span>, stores_grad=<span class="hljs-literal">True</span>)
w0.gaussian(<span class="hljs-number">0.0</span>, <span class="hljs-number">0.1</span>)
b0 = Tensor(shape=(<span class="hljs-number">1</span>, <span class="hljs-number">3</span>), requires_grad=<span class="hljs-literal">True</span>, stores_grad=<span class="hljs-literal">True</span>)
b0.set_value(<span class="hljs-number">0.0</span>)

w1 = Tensor(shape=(<span class="hljs-number">3</span>, <span class="hljs-number">2</span>), requires_grad=<span class="hljs-literal">True</span>, stores_grad=<span class="hljs-literal">True</span>)
w1.gaussian(<span class="hljs-number">0.0</span>, <span class="hljs-number">0.1</span>)
b1 = Tensor(shape=(<span class="hljs-number">1</span>, <span class="hljs-number">2</span>), requires_grad=<span class="hljs-literal">True</span>, stores_grad=<span class="hljs-literal">True</span>)
b1.set_value(<span class="hljs-number">0.0</span>)
</code></pre>
<h4><a class="anchor" aria-hidden="true" id="training"></a><a href="#training" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Training</h4>
<pre><code class="hljs css language-python">inputs = Tensor(data=data)  <span class="hljs-comment"># data matrix</span>
target = Tensor(data=label) <span class="hljs-comment"># label vector</span>
autograd.training = <span class="hljs-literal">True</span>    <span class="hljs-comment"># for training</span>
sgd = opt.SGD(<span class="hljs-number">0.05</span>)   <span class="hljs-comment"># optimizer</span>

<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(<span class="hljs-number">10</span>):
    x = autograd.matmul(inputs, w0) <span class="hljs-comment"># matrix multiplication</span>
    x = autograd.add_bias(x, b0)    <span class="hljs-comment"># add the bias vector</span>
    x = autograd.relu(x)            <span class="hljs-comment"># ReLU activation operation</span>

    x = autograd.matmul(x, w1)
    x = autograd.add_bias(x, b1)

    loss = autograd.softmax_cross_entropy(x, target)

    <span class="hljs-keyword">for</span> p, g <span class="hljs-keyword">in</span> autograd.backward(loss):
        sgd.update(p, g)
</code></pre>
<h3><a class="anchor" aria-hidden="true" id="operation--layer"></a><a href="#operation--layer" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Operation + Layer</h3>
<p>The following <a href="https://github.com/apache/singa/blob/master/examples/autograd/mnist_cnn.py">example</a> implements a CNN model using layers provided by the autograd module.</p>
<h4><a class="anchor" aria-hidden="true" id="create-the-layers"></a><a href="#create-the-layers" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Create the layers</h4>
<pre><code class="hljs css language-python">conv1 = autograd.Conv2d(<span class="hljs-number">1</span>, <span class="hljs-number">32</span>, <span class="hljs-number">3</span>, padding=<span class="hljs-number">1</span>, bias=<span class="hljs-literal">False</span>)
bn1 = autograd.BatchNorm2d(<span class="hljs-number">32</span>)
pooling1 = autograd.MaxPool2d(<span class="hljs-number">3</span>, <span class="hljs-number">1</span>, padding=<span class="hljs-number">1</span>)
conv21 = autograd.Conv2d(<span class="hljs-number">32</span>, <span class="hljs-number">16</span>, <span class="hljs-number">3</span>, padding=<span class="hljs-number">1</span>)
conv22 = autograd.Conv2d(<span class="hljs-number">32</span>, <span class="hljs-number">16</span>, <span class="hljs-number">3</span>, padding=<span class="hljs-number">1</span>)
bn2 = autograd.BatchNorm2d(<span class="hljs-number">32</span>)
linear = autograd.Linear(<span class="hljs-number">32</span> * <span class="hljs-number">28</span> * <span class="hljs-number">28</span>, <span class="hljs-number">10</span>)
pooling2 = autograd.AvgPool2d(<span class="hljs-number">3</span>, <span class="hljs-number">1</span>, padding=<span class="hljs-number">1</span>)
</code></pre>
<h4><a class="anchor" aria-hidden="true" id="define-the-forward-function"></a><a href="#define-the-forward-function" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Define the forward function</h4>
<p>The operations in the forward pass will be recorded automatically for backward propagation.</p>
<pre><code class="hljs css language-python"><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">forward</span><span class="hljs-params">(x, t)</span>:</span>
    <span class="hljs-comment"># x is the input data (a batch of images)</span>
    <span class="hljs-comment"># t the the label vector (a batch of integers)</span>
    y = conv1(x)           <span class="hljs-comment"># Conv layer</span>
    y = autograd.relu(y)   <span class="hljs-comment"># ReLU operation</span>
    y = bn1(y)             <span class="hljs-comment"># BN layer</span>
    y = pooling1(y)        <span class="hljs-comment"># Pooling Layer</span>

    <span class="hljs-comment"># two parallel convolution layers</span>
    y1 = conv21(y)
    y2 = conv22(y)
    y = autograd.cat((y1, y2), <span class="hljs-number">1</span>)  <span class="hljs-comment"># cat operation</span>
    y = autograd.relu(y)           <span class="hljs-comment"># ReLU operation</span>
    y = bn2(y)
    y = pooling2(y)

    y = autograd.flatten(y)        <span class="hljs-comment"># flatten operation</span>
    y = linear(y)                  <span class="hljs-comment"># Linear layer</span>
    loss = autograd.softmax_cross_entropy(y, t)  <span class="hljs-comment"># operation</span>
    <span class="hljs-keyword">return</span> loss, y
</code></pre>
<h4><a class="anchor" aria-hidden="true" id="training-1"></a><a href="#training-1" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Training</h4>
<pre><code class="hljs css language-python">autograd.training = <span class="hljs-literal">True</span>
<span class="hljs-keyword">for</span> epoch <span class="hljs-keyword">in</span> range(epochs):
    <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(batch_number):
        inputs = tensor.Tensor(device=dev, data=x_train[
                               i * batch_sz:(<span class="hljs-number">1</span> + i) * batch_sz], stores_grad=<span class="hljs-literal">False</span>)
        targets = tensor.Tensor(device=dev, data=y_train[
                                i * batch_sz:(<span class="hljs-number">1</span> + i) * batch_sz], requires_grad=<span class="hljs-literal">False</span>, stores_grad=<span class="hljs-literal">False</span>)

        loss, y = forward(inputs, targets) <span class="hljs-comment"># forward the net</span>

        <span class="hljs-keyword">for</span> p, gp <span class="hljs-keyword">in</span> autograd.backward(loss):  <span class="hljs-comment"># auto backward</span>
            sgd.update(p, gp)
</code></pre>
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