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<h1>Graph Attention Networks v2 (GATv2)</h1>
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<p>This is a <a href="https://pytorch.org">PyTorch</a> implementation of the GATv2 operator from the paper
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<a href="https://arxiv.org/abs/2105.14491">How Attentive are Graph Attention Networks?</a>.</p>
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<p>GATv2s work on graph data similar to <a href="../gat/index.html">GAT</a>.
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A graph consists of nodes and edges connecting nodes.
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For example, in Cora dataset the nodes are research papers and the edges are citations that
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connect the papers.</p>
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<p>The GATv2 operator fixes the static attention problem of the standard <a href="../gat/index.html">GAT</a>.
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Static attention is when the attention to the key nodes has the same rank (order) for any query node.
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<a href="../gat/index.html">GAT</a> computes attention from query node $i$ to key node $j$ as,</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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e_{ij} &= \text{LeakyReLU} \Big(\mathbf{a}^\top \Big[
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\mathbf{W} \overrightarrow{h_i} \Vert \mathbf{W} \overrightarrow{h_j}
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\Big] \Big) \\
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&=
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\text{LeakyReLU} \Big(\mathbf{a}_1^\top \mathbf{W} \overrightarrow{h_i} +
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\mathbf{a}_2^\top \mathbf{W} \overrightarrow{h_j}
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\Big)
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\end{align}</script>
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</p>
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<p>Note that for any query node $i$, the attention rank ($argsort$) of keys depends only
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on $\mathbf{a}_2^\top \mathbf{W} \overrightarrow{h_j}$.
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Therefore the attention rank of keys remains the same (<em>static</em>) for all queries.</p>
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<p>GATv2 allows dynamic attention by changing the attention mechanism,</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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e_{ij} &= \mathbf{a}^\top \text{LeakyReLU} \Big( \mathbf{W} \Big[
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\overrightarrow{h_i} \Vert \overrightarrow{h_j}
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\Big] \Big) \\
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&= \mathbf{a}^\top \text{LeakyReLU} \Big(
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\mathbf{W}_l \overrightarrow{h_i} + \mathbf{W}_r \overrightarrow{h_j}
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\Big)
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\end{align}</script>
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</p>
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<p>The paper shows that GATs static attention mechanism fails on some graph problems
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with a synthetic dictionary lookup dataset.
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It’s a fully connected bipartite graph where one set of nodes (query nodes)
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have a key associated with it
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and the other set of nodes have both a key and a value associated with it.
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The goal is to predict the values of query nodes.
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GAT fails on this task because of its limited static attention.</p>
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<p>Here is <a href="experiment.html">the training code</a> for training
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a two-layer GATv2 on Cora dataset.</p>
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<p><a href="https://app.labml.ai/run/34b1e2f6ed6f11ebb860997901a2d1e3"><img alt="View Run" src="https://img.shields.io/badge/labml-experiment-brightgreen" /></a></p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">59</span><span></span><span class="kn">import</span> <span class="nn">torch</span>
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<span class="lineno">60</span><span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
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<span class="lineno">61</span>
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<span class="lineno">62</span><span class="kn">from</span> <span class="nn">labml_helpers.module</span> <span class="kn">import</span> <span class="n">Module</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-1'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-1'>#</a>
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</div>
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<h2>Graph attention v2 layer</h2>
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<p>This is a single graph attention v2 layer.
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A GATv2 is made up of multiple such layers.
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It takes
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<script type="math/tex; mode=display">\mathbf{h} = \{ \overrightarrow{h_1}, \overrightarrow{h_2}, \dots, \overrightarrow{h_N} \}</script>,
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where $\overrightarrow{h_i} \in \mathbb{R}^F$ as input
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and outputs
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<script type="math/tex; mode=display">\mathbf{h'} = \{ \overrightarrow{h'_1}, \overrightarrow{h'_2}, \dots, \overrightarrow{h'_N} \}</script>,
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where $\overrightarrow{h’_i} \in \mathbb{R}^{F’}$.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">65</span><span class="k">class</span> <span class="nc">GraphAttentionV2Layer</span><span class="p">(</span><span class="n">Module</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-2'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-2'>#</a>
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</div>
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<ul>
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<li><code>in_features</code>, $F$, is the number of input features per node</li>
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<li><code>out_features</code>, $F’$, is the number of output features per node</li>
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<li><code>n_heads</code>, $K$, is the number of attention heads</li>
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<li><code>is_concat</code> whether the multi-head results should be concatenated or averaged</li>
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<li><code>dropout</code> is the dropout probability</li>
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<li><code>leaky_relu_negative_slope</code> is the negative slope for leaky relu activation</li>
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<li><code>share_weights</code> if set to <code>True</code>, the same matrix will be applied to the source and the target node of every edge</li>
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</ul>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">78</span> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">in_features</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">out_features</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">n_heads</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
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<span class="lineno">79</span> <span class="n">is_concat</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="kc">True</span><span class="p">,</span>
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<span class="lineno">80</span> <span class="n">dropout</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">0.6</span><span class="p">,</span>
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<span class="lineno">81</span> <span class="n">leaky_relu_negative_slope</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">0.2</span><span class="p">,</span>
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<span class="lineno">82</span> <span class="n">share_weights</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="kc">False</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-3'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-3'>#</a>
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</div>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">92</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
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<span class="lineno">93</span>
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<span class="lineno">94</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_concat</span> <span class="o">=</span> <span class="n">is_concat</span>
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<span class="lineno">95</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_heads</span> <span class="o">=</span> <span class="n">n_heads</span>
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<span class="lineno">96</span> <span class="bp">self</span><span class="o">.</span><span class="n">share_weights</span> <span class="o">=</span> <span class="n">share_weights</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-4'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-4'>#</a>
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</div>
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<p>Calculate the number of dimensions per head</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">99</span> <span class="k">if</span> <span class="n">is_concat</span><span class="p">:</span>
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<span class="lineno">100</span> <span class="k">assert</span> <span class="n">out_features</span> <span class="o">%</span> <span class="n">n_heads</span> <span class="o">==</span> <span class="mi">0</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-5'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-5'>#</a>
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</div>
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<p>If we are concatenating the multiple heads</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">102</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span> <span class="o">=</span> <span class="n">out_features</span> <span class="o">//</span> <span class="n">n_heads</span>
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<span class="lineno">103</span> <span class="k">else</span><span class="p">:</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-6'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-6'>#</a>
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</div>
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<p>If we are averaging the multiple heads</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">105</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span> <span class="o">=</span> <span class="n">out_features</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-7'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-7'>#</a>
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</div>
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<p>Linear layer for initial source transformation;
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i.e. to transform the source node embeddings before self-attention</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">109</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear_l</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="n">in_features</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span> <span class="o">*</span> <span class="n">n_heads</span><span class="p">,</span> <span class="n">bias</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-8'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-8'>#</a>
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</div>
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<p>If <code>share_weights</code> is <code>True</code> the same linear layer is used for the target nodes</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">111</span> <span class="k">if</span> <span class="n">share_weights</span><span class="p">:</span>
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<span class="lineno">112</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear_r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear_l</span>
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<span class="lineno">113</span> <span class="k">else</span><span class="p">:</span>
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<span class="lineno">114</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear_r</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="n">in_features</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span> <span class="o">*</span> <span class="n">n_heads</span><span class="p">,</span> <span class="n">bias</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-9'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-9'>#</a>
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</div>
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<p>Linear layer to compute attention score $e_{ij}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">116</span> <span class="bp">self</span><span class="o">.</span><span class="n">attn</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">bias</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-10'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-10'>#</a>
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</div>
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<p>The activation for attention score $e_{ij}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">118</span> <span class="bp">self</span><span class="o">.</span><span class="n">activation</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">LeakyReLU</span><span class="p">(</span><span class="n">negative_slope</span><span class="o">=</span><span class="n">leaky_relu_negative_slope</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-11'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-11'>#</a>
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</div>
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<p>Softmax to compute attention $\alpha_{ij}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">120</span> <span class="bp">self</span><span class="o">.</span><span class="n">softmax</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Softmax</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-12'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-12'>#</a>
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</div>
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<p>Dropout layer to be applied for attention</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">122</span> <span class="bp">self</span><span class="o">.</span><span class="n">dropout</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Dropout</span><span class="p">(</span><span class="n">dropout</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-13'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-13'>#</a>
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</div>
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<ul>
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<li><code>h</code>, $\mathbf{h}$ is the input node embeddings of shape <code>[n_nodes, in_features]</code>.</li>
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<li><code>adj_mat</code> is the adjacency matrix of shape <code>[n_nodes, n_nodes, n_heads]</code>.
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We use shape <code>[n_nodes, n_nodes, 1]</code> since the adjacency is the same for each head.
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Adjacency matrix represent the edges (or connections) among nodes.
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<code>adj_mat[i][j]</code> is <code>True</code> if there is an edge from node <code>i</code> to node <code>j</code>.</li>
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</ul>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">124</span> <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">h</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">adj_mat</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span></pre></div>
|
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</div>
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</div>
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|
<div class='section' id='section-14'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
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|
<a href='#section-14'>#</a>
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|
</div>
|
|
<p>Number of nodes</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">134</span> <span class="n">n_nodes</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span></pre></div>
|
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</div>
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</div>
|
|
<div class='section' id='section-15'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
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|
<a href='#section-15'>#</a>
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|
</div>
|
|
<p>The initial transformations,
|
|
<script type="math/tex; mode=display">\overrightarrow{{g_l}^k_i} = \mathbf{W_l}^k \overrightarrow{h_i}</script>
|
|
<script type="math/tex; mode=display">\overrightarrow{{g_r}^k_i} = \mathbf{W_r}^k \overrightarrow{h_i}</script>
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|
for each head.
|
|
We do two linear transformations and then split it up for each head.</p>
|
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</div>
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<div class='code'>
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|
<div class="highlight"><pre><span class="lineno">140</span> <span class="n">g_l</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear_l</span><span class="p">(</span><span class="n">h</span><span class="p">)</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">n_nodes</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_heads</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span><span class="p">)</span>
|
|
<span class="lineno">141</span> <span class="n">g_r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear_r</span><span class="p">(</span><span class="n">h</span><span class="p">)</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">n_nodes</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_heads</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-16'>
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|
<div class='docs'>
|
|
<div class='section-link'>
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|
<a href='#section-16'>#</a>
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|
</div>
|
|
<h4>Calculate attention score</h4>
|
|
<p>We calculate these for each head $k$. <em>We have omitted $\cdot^k$ for simplicity</em>.</p>
|
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<p>
|
|
<script type="math/tex; mode=display">e_{ij} = a(\mathbf{W_l} \overrightarrow{h_i}, \mathbf{W_r} \overrightarrow{h_j}) =
|
|
a(\overrightarrow{{g_l}_i}, \overrightarrow{{g_r}_j})</script>
|
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</p>
|
|
<p>$e_{ij}$ is the attention score (importance) from node $j$ to node $i$.
|
|
We calculate this for each head.</p>
|
|
<p>$a$ is the attention mechanism, that calculates the attention score.
|
|
The paper sums
|
|
$\overrightarrow{{g_l}_i}$, $\overrightarrow{{g_r}_j}$
|
|
followed by a $\text{LeakyReLU}$
|
|
and does a linear transformation with a weight vector $\mathbf{a} \in \mathbb{R}^{F’}$</p>
|
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<p>
|
|
<script type="math/tex; mode=display">e_{ij} = \mathbf{a}^\top \text{LeakyReLU} \Big(
|
|
\Big[
|
|
\overrightarrow{{g_l}_i} + \overrightarrow{{g_r}_j}
|
|
\Big] \Big)</script>
|
|
Note: The paper desrcibes $e_{ij}$ as <br />
|
|
<script type="math/tex; mode=display">e_{ij} = \mathbf{a}^\top \text{LeakyReLU} \Big( \mathbf{W}
|
|
\Big[
|
|
\overrightarrow{h_i} \Vert \overrightarrow{h_j}
|
|
\Big] \Big)</script>
|
|
which is equivalent to the definition we use here.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-17'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-17'>#</a>
|
|
</div>
|
|
<p>First we calculate
|
|
$\Big[\overrightarrow{{g_l}_i} + \overrightarrow{{g_r}_j} \Big]$
|
|
for all pairs of $i, j$.</p>
|
|
<p><code>g_l_repeat</code> gets
|
|
<script type="math/tex; mode=display">\{\overrightarrow{{g_l}_1}, \overrightarrow{{g_l}_2}, \dots, \overrightarrow{{g_l}_N},
|
|
\overrightarrow{{g_l}_1}, \overrightarrow{{g_l}_2}, \dots, \overrightarrow{{g_l}_N}, ...\}</script>
|
|
where each node embedding is repeated <code>n_nodes</code> times.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">179</span> <span class="n">g_l_repeat</span> <span class="o">=</span> <span class="n">g_l</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">n_nodes</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-18'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-18'>#</a>
|
|
</div>
|
|
<p><code>g_r_repeat_interleave</code> gets
|
|
<script type="math/tex; mode=display">\{\overrightarrow{{g_r}_1}, \overrightarrow{{g_r}_1}, \dots, \overrightarrow{{g_r}_1},
|
|
\overrightarrow{{g_r}_2}, \overrightarrow{{g_r}_2}, \dots, \overrightarrow{{g_r}_2}, ...\}</script>
|
|
where each node embedding is repeated <code>n_nodes</code> times.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">184</span> <span class="n">g_r_repeat_interleave</span> <span class="o">=</span> <span class="n">g_r</span><span class="o">.</span><span class="n">repeat_interleave</span><span class="p">(</span><span class="n">n_nodes</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-19'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-19'>#</a>
|
|
</div>
|
|
<p>Now we add the two tensors to get
|
|
<script type="math/tex; mode=display">\{\overrightarrow{{g_l}_1} + \overrightarrow{{g_r}_1},
|
|
\overrightarrow{{g_l}_1}, + \overrightarrow{{g_r}_2},
|
|
\dots, \overrightarrow{{g_l}_1} +\overrightarrow{{g_r}_N},
|
|
\overrightarrow{{g_l}_2} + \overrightarrow{{g_r}_1},
|
|
\overrightarrow{{g_l}_2}, + \overrightarrow{{g_r}_2},
|
|
\dots, \overrightarrow{{g_l}_2} + \overrightarrow{{g_r}_N}, ...\}</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">192</span> <span class="n">g_sum</span> <span class="o">=</span> <span class="n">g_l_repeat</span> <span class="o">+</span> <span class="n">g_r_repeat_interleave</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-20'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-20'>#</a>
|
|
</div>
|
|
<p>Reshape so that <code>g_sum[i, j]</code> is $\overrightarrow{{g_l}_i} + \overrightarrow{{g_r}_j}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">194</span> <span class="n">g_sum</span> <span class="o">=</span> <span class="n">g_sum</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">n_nodes</span><span class="p">,</span> <span class="n">n_nodes</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_heads</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-21'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-21'>#</a>
|
|
</div>
|
|
<p>Calculate
|
|
<script type="math/tex; mode=display">e_{ij} = \mathbf{a}^\top \text{LeakyReLU} \Big(
|
|
\Big[
|
|
\overrightarrow{{g_l}_i} + \overrightarrow{{g_r}_j}
|
|
\Big] \Big)</script>
|
|
<code>e</code> is of shape <code>[n_nodes, n_nodes, n_heads, 1]</code></p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">202</span> <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">attn</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">activation</span><span class="p">(</span><span class="n">g_sum</span><span class="p">))</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-22'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-22'>#</a>
|
|
</div>
|
|
<p>Remove the last dimension of size <code>1</code></p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">204</span> <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-23'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-23'>#</a>
|
|
</div>
|
|
<p>The adjacency matrix should have shape
|
|
<code>[n_nodes, n_nodes, n_heads]</code> or<code>[n_nodes, n_nodes, 1]</code></p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">208</span> <span class="k">assert</span> <span class="n">adj_mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">adj_mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">n_nodes</span>
|
|
<span class="lineno">209</span> <span class="k">assert</span> <span class="n">adj_mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">adj_mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">n_nodes</span>
|
|
<span class="lineno">210</span> <span class="k">assert</span> <span class="n">adj_mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">adj_mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_heads</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-24'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-24'>#</a>
|
|
</div>
|
|
<p>Mask $e_{ij}$ based on adjacency matrix.
|
|
$e_{ij}$ is set to $- \infty$ if there is no edge from $i$ to $j$.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">213</span> <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">masked_fill</span><span class="p">(</span><span class="n">adj_mat</span> <span class="o">==</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">float</span><span class="p">(</span><span class="s1">'-inf'</span><span class="p">))</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-25'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-25'>#</a>
|
|
</div>
|
|
<p>We then normalize attention scores (or coefficients)
|
|
<script type="math/tex; mode=display">\alpha_{ij} = \text{softmax}_j(e_{ij}) =
|
|
\frac{\exp(e_{ij})}{\sum_{j \in \mathcal{N}_i} \exp(e_{ij})}</script>
|
|
</p>
|
|
<p>where $\mathcal{N}_i$ is the set of nodes connected to $i$.</p>
|
|
<p>We do this by setting unconnected $e_{ij}$ to $- \infty$ which
|
|
makes $\exp(e_{ij}) \sim 0$ for unconnected pairs.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">223</span> <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">softmax</span><span class="p">(</span><span class="n">e</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-26'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-26'>#</a>
|
|
</div>
|
|
<p>Apply dropout regularization</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">226</span> <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dropout</span><span class="p">(</span><span class="n">a</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-27'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-27'>#</a>
|
|
</div>
|
|
<p>Calculate final output for each head
|
|
<script type="math/tex; mode=display">\overrightarrow{h'^k_i} = \sum_{j \in \mathcal{N}_i} \alpha^k_{ij} \overrightarrow{{g_r}_{j,k}}</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">230</span> <span class="n">attn_res</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'ijh,jhf->ihf'</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">g_r</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-28'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-28'>#</a>
|
|
</div>
|
|
<p>Concatenate the heads</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">233</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_concat</span><span class="p">:</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-29'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-29'>#</a>
|
|
</div>
|
|
<p>
|
|
<script type="math/tex; mode=display">\overrightarrow{h'_i} = \Bigg\Vert_{k=1}^{K} \overrightarrow{h'^k_i}</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">235</span> <span class="k">return</span> <span class="n">attn_res</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n_nodes</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_heads</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-30'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-30'>#</a>
|
|
</div>
|
|
<p>Take the mean of the heads</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">237</span> <span class="k">else</span><span class="p">:</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-31'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-31'>#</a>
|
|
</div>
|
|
<p>
|
|
<script type="math/tex; mode=display">\overrightarrow{h'_i} = \frac{1}{K} \sum_{k=1}^{K} \overrightarrow{h'^k_i}</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">239</span> <span class="k">return</span> <span class="n">attn_res</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js?config=TeX-AMS_HTML">
|
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</script>
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<!-- MathJax configuration -->
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<script type="text/x-mathjax-config">
|
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MathJax.Hub.Config({
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tex2jax: {
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inlineMath: [ ['$','$'] ],
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// Center justify equations in code and markdown cells. Elsewhere
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// we use CSS to left justify single line equations in code cells.
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displayAlign: 'center',
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"HTML-CSS": { fonts: ["TeX"] }
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});
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</script>
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<script>
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img.parentElement.style.textAlign = 'center'
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modal.id = 'modal'
|
|
|
|
var modalContent = document.createElement('div')
|
|
modal.appendChild(modalContent)
|
|
|
|
var modalImage = document.createElement('img')
|
|
modalContent.appendChild(modalImage)
|
|
|
|
var span = document.createElement('span')
|
|
span.classList.add('close')
|
|
span.textContent = 'x'
|
|
modal.appendChild(span)
|
|
|
|
img.onclick = function () {
|
|
console.log('clicked')
|
|
document.body.appendChild(modal)
|
|
modalImage.src = img.src
|
|
}
|
|
|
|
span.onclick = function () {
|
|
document.body.removeChild(modal)
|
|
}
|
|
}
|
|
|
|
handleImages()
|
|
</script>
|
|
</body>
|
|
</html> |