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<a class="parent" href="/">home</a>
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<a class="parent" href="../index.html">transformers</a>
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<a class="parent" href="index.html">aft</a>
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<div class='section' id='section-0'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-0'>#</a>
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</div>
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<h1>An Attention Free Transformer</h1>
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<p>This is a <a href="https://pytorch.org">PyTorch</a> implementation of the paper
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<a href="https://papers.labml.ai/paper/2105.14103">An Attention Free Transformer</a>.</p>
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<p>This paper replaces the <a href="../mha.html">self-attention layer</a> with a new efficient operation,
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that has memory complexity of $\mathcal{O}(Td)$, where $T$ is the sequence length
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and $d$ is the dimensionality of embeddings.</p>
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<p>The paper introduces AFT along with AFT-local and AFT-conv.
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Here we have implemented AFT-local which pays attention to closeby tokens
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in an autoregressive model.</p>
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<h2>Attention Free Transformer</h2>
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<p>AFT (similar to <a href="../mha.html">MHA</a>) first transforms the embeddings $X$ into
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query $Q = XW^Q$, key $K = XW^K$ and value $V = XW^V$ tensors with learned weights.
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The output for each position $t \in [1, T]$ is calculated with the following operation.</p>
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<p>
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<script type="math/tex; mode=display">Y_t = \sigma(Q_t) \odot
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\frac{\sum_{t'=1}^T \exp(K_{t'} + w_{t,t'}) \odot V_{t'}}
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{\sum_{t'=1}^T \exp(K_{t'} + w_{t,t'})}</script>
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</p>
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<p>, where $\odot$ is element-wise product, $\sigma$ is a non-linearity (sigmoid) and
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$w \in \mathbb{R}^{T \times T}$ is a learned matrix of pair-wise position biases.</p>
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<p>This means that we take the weighted average of values
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and multiply them by the query. This eliminates the need to calculate the $T \times T$ attention
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matrix that <a href="../mha.html">MHA</a> requires, and therefore reduce the memory requirement.</p>
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<h2>AFT Local</h2>
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<p>AFT Local only apply learned pair-wise position biases locally:</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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w'_{t,t'} =
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\begin{cases}
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w_{t,t'}, & \text{for $\lvert t-t' \rvert \lt s$} \\
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0, & \text{otherwise}
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\end{cases}
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\end{align}</script>
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</p>
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<p>, where $s \le T$ is the local window size.</p>
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<p>Although $w’_{t,t’}$ is $0$ outside the local window the AFT operation still uses key-value pairs from
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other areas. This is different from local transformers where embeddings outside the local window are
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completely not visible.</p>
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<p>Here is <a href="experiment.html">the training code</a> for a AFT Local model.</p>
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<p><a href="https://app.labml.ai/run/6348e504c3a511eba9529daa283fb495"><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">61</span><span></span><span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Optional</span>
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<span class="lineno">62</span>
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<span class="lineno">63</span><span class="kn">import</span> <span class="nn">torch</span>
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<span class="lineno">64</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">65</span>
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<span class="lineno">66</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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<h3>AFT Local Operation</h3>
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<p>
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<script type="math/tex; mode=display">Y_t = \sigma(Q_t) \odot
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\frac{\sum_{t'=1}^T \exp(K_{t'} + w_{t,t'}) \odot V_{t'}}
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{\sum_{t'=1}^T \exp(K_{t'} + w_{t,t'})}</script>
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</p>
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<p>where,</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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w'_{t,t'} =
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\begin{cases}
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w_{t,t'}, & \text{for $\lvert t-t' \rvert \lt s$} \\
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0, & \text{otherwise}
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\end{cases}
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\end{align}</script>
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</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">69</span><span class="k">class</span> <span class="nc">AFTLocal</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>d_model</code> is the number of features in the <code>query</code>, <code>key</code> and <code>value</code> vectors.</li>
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<li><code>seq_len</code> is $T$</li>
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<li><code>local_window_size</code> is the local window size $s$</li>
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<li><code>bias</code> is whether to have a bias parameter for transformations for $Q$, $K$ and $V$.</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">88</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">d_model</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">seq_len</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">local_window_size</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">bias</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="kc">True</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">96</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</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-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>Local window size $s$</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="bp">self</span><span class="o">.</span><span class="n">local_window_size</span> <span class="o">=</span> <span class="n">local_window_size</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>These transform the <code>query</code>, <code>key</code> and <code>value</code> vectors.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">101</span> <span class="bp">self</span><span class="o">.</span><span class="n">query</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">d_model</span><span class="p">,</span> <span class="n">d_model</span><span class="p">,</span> <span class="n">bias</span><span class="o">=</span><span class="n">bias</span><span class="p">)</span>
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<span class="lineno">102</span> <span class="bp">self</span><span class="o">.</span><span class="n">key</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">d_model</span><span class="p">,</span> <span class="n">d_model</span><span class="p">,</span> <span class="n">bias</span><span class="o">=</span><span class="n">bias</span><span class="p">)</span>
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<span class="lineno">103</span> <span class="bp">self</span><span class="o">.</span><span class="n">value</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">d_model</span><span class="p">,</span> <span class="n">d_model</span><span class="p">,</span> <span class="n">bias</span><span class="o">=</span><span class="n">bias</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>Pair-wise positional biases $w \in \mathbb{R}^{T \times T}$</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">pos_bias</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">seq_len</span><span class="p">,</span> <span class="n">seq_len</span><span class="p">),</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</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-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>Mask for $w_{t,t’}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">107</span> <span class="bp">self</span><span class="o">.</span><span class="n">local_mask</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">create_local_mask</span><span class="p">(</span><span class="n">seq_len</span><span class="p">,</span> <span class="n">local_window_size</span><span class="p">),</span> <span class="n">requires_grad</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>Activation $\sigma$</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">activation</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Sigmoid</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>Output layer</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="bp">self</span><span class="o">.</span><span class="n">output</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">d_model</span><span class="p">,</span> <span class="n">d_model</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 doc-strings'>
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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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<h4>Create local mask</h4>
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<p>This creates a mask for
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<script type="math/tex; mode=display">\begin{align}
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m_{t,t'} =
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\begin{cases}
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1, & \text{for $\lvert t-t' \rvert \lt s$} \\
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0, & \text{otherwise}
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\end{cases}
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\end{align}</script>
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</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">113</span> <span class="nd">@staticmethod</span>
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<span class="lineno">114</span> <span class="k">def</span> <span class="nf">create_local_mask</span><span class="p">(</span><span class="n">seq_len</span><span class="p">,</span> <span class="n">local_window_size</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>Initialize to ones</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">129</span> <span class="n">local_mask</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">seq_len</span><span class="p">,</span> <span class="n">seq_len</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">bool</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>Make $t’ - t \ge s$ zero</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">131</span> <span class="n">local_mask</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tril</span><span class="p">(</span><span class="n">local_mask</span><span class="p">,</span> <span class="n">local_window_size</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-13'>
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<div class='docs'>
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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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<p>Make $t - t’ \ge s$ zero</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">133</span> <span class="n">local_mask</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">triu</span><span class="p">(</span><span class="n">local_mask</span><span class="p">,</span> <span class="o">-</span><span class="p">(</span><span class="n">local_window_size</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-14'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-14'>#</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">136</span> <span class="k">return</span> <span class="n">local_mask</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-15'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-15'>#</a>
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</div>
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<p><code>query</code>, <code>key</code> and <code>value</code> are the tensors that store
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collection of token embeddings for <em>query</em>, <em>key</em> and <em>value</em>.
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They have shape <code>[seq_len, batch_size, d_model]</code>.</p>
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<p><code>mask</code> has shape <code>[seq_len, seq_len, batch_size]</code> and
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<code>mask[i, j, b]</code> indicates whether for batch <code>b</code>,
|
|
query at position <code>i</code> has access to key-value at position <code>j</code>.</p>
|
|
</div>
|
|
<div class='code'>
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|
<div class="highlight"><pre><span class="lineno">138</span> <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="p">,</span>
|
|
<span class="lineno">139</span> <span class="n">query</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="lineno">140</span> <span class="n">key</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="lineno">141</span> <span class="n">value</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="lineno">142</span> <span class="n">mask</span><span class="p">:</span> <span class="n">Optional</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="o">=</span> <span class="kc">None</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>
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<p><code>query</code>, <code>key</code> and <code>value</code> have shape <code>[seq_len, batch_size, d_model]</code></p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">154</span> <span class="n">seq_len</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">query</span><span class="o">.</span><span class="n">shape</span>
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<span class="lineno">155</span>
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<span class="lineno">156</span> <span class="k">if</span> <span class="n">mask</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</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-17'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-17'>#</a>
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</div>
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<p><code>mask</code> has shape <code>[seq_len_q, seq_len_k, batch_size]</code>,
|
|
where first dimension is the query dimension.
|
|
If the query dimension is equal to $1$ it will be broadcasted.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">160</span> <span class="k">assert</span> <span class="n">mask</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">mask</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">query</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="lineno">161</span> <span class="k">assert</span> <span class="n">mask</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">key</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="lineno">162</span> <span class="k">assert</span> <span class="n">mask</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">mask</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="n">query</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span></pre></div>
|
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</div>
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|
</div>
|
|
<div class='section' id='section-18'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-18'>#</a>
|
|
</div>
|
|
<p>Transform query, key and value embeddings</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">165</span> <span class="n">query</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">query</span><span class="p">(</span><span class="n">query</span><span class="p">)</span>
|
|
<span class="lineno">166</span> <span class="n">key</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">key</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
|
|
<span class="lineno">167</span> <span class="n">value</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">value</span><span class="p">(</span><span class="n">value</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>Get
|
|
<script type="math/tex; mode=display">\begin{align}
|
|
w'_{t,t'} =
|
|
\begin{cases}
|
|
w_{t,t'}, & \text{for $\lvert t-t' \rvert \lt s$} \\
|
|
0, & \text{otherwise}
|
|
\end{cases}
|
|
\end{align}</script>
|
|
using the mask</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">178</span> <span class="n">pos_bias</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">pos_bias</span><span class="p">[:</span><span class="n">seq_len</span><span class="p">,</span> <span class="p">:</span><span class="n">seq_len</span><span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">local_mask</span><span class="p">[:</span><span class="n">seq_len</span><span class="p">,</span> <span class="p">:</span><span class="n">seq_len</span><span class="p">]</span>
|
|
<span class="lineno">179</span> <span class="n">pos_bias</span> <span class="o">=</span> <span class="n">pos_bias</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
|
<span class="lineno">180</span> <span class="n">pos_bias</span><span class="o">.</span><span class="n">masked_fill_</span><span class="p">(</span><span class="o">~</span><span class="n">mask</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-20'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-20'>#</a>
|
|
</div>
|
|
<p>
|
|
<script type="math/tex; mode=display">\begin{align}
|
|
Y_t &= \sigma(Q_t) \odot
|
|
\frac{\sum_{t'=1}^T \exp(K_{t'} + w_{t,t'}) \odot V_{t'}}
|
|
{\sum_{t'=1}^T \exp(K_{t'} + w_{t,t'})} \\
|
|
&= \sigma(Q_t) \odot
|
|
\frac{\sum_{t'=1}^T \exp(w_{t,t'}) \odot \exp(K_{t'}) \odot V_{t'}}
|
|
{\sum_{t'=1}^T \exp(w_{t,t'}) \odot \exp(K_{t'})}
|
|
\end{align}</script>
|
|
</p>
|
|
<p>We compute $\exp(w_{t,t’})$, $\exp(K_{t’}) \odot V_{t’}$ and $\exp(K_{t’})$
|
|
separately and do a matrix multiplication. We use einsum for clarity.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-21'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-21'>#</a>
|
|
</div>
|
|
<p>We subtract $\max_{t’}(K_{t’})$ and $\max_{t’}(w_{t,t’})$ before calculating the exponents to stabilize
|
|
the softmax calculation.</p>
|
|
<p>If $x_i$ is large $\exp(x_i)$ becomes huge and the computation of
|
|
$\frac{\sum\exp(x_i)y_i}{\sum\exp(x_i)}$becomes unstable.
|
|
Subtracting a constant before calculating the exponent from numerator and denominator will cancel out.
|
|
and can help stabilize the computation.
|
|
So we subtract $\max(x_i)$ to stabilize the computation.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">202</span> <span class="n">max_key</span> <span class="o">=</span> <span class="n">key</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
|
|
<span class="lineno">203</span> <span class="n">max_pos_bias</span> <span class="o">=</span> <span class="n">pos_bias</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)[</span><span class="mi">0</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>$\exp \big(K_{t’}- \max_{t’}(K_{t’})\big)$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">206</span> <span class="n">exp_key</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">key</span> <span class="o">-</span> <span class="n">max_key</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>$\exp \big(w_{t,t’} - \max_{t’}(w_{t,t’})\big)$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">208</span> <span class="n">exp_pos_bias</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">pos_bias</span> <span class="o">-</span> <span class="n">max_pos_bias</span><span class="p">)</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>The numerator part $\sum_{t’=1}^T \exp(w_{t,t’}) \odot \exp(K_{t’}) \odot V_{t’}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">211</span> <span class="n">num</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">'ijb,jbd->ibd'</span><span class="p">,</span> <span class="n">exp_pos_bias</span><span class="p">,</span> <span class="n">exp_key</span> <span class="o">*</span> <span class="n">value</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>The denominator part $\sum_{t’=1}^T \exp(w_{t,t’}) \odot \exp(K_{t’})$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">213</span> <span class="n">den</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">'ijb,jbd->ibd'</span><span class="p">,</span> <span class="n">exp_pos_bias</span><span class="p">,</span> <span class="n">exp_key</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>Output <script type="math/tex; mode=display">Y_t = \sigma(Q_t) \odot
|
|
\frac{\sum_{t'=1}^T \exp(w_{t,t'}) \odot \exp(K_{t'}) \odot V_{t'}}
|
|
{\sum_{t'=1}^T \exp(w_{t,t'}) \odot \exp(K_{t'})}</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">218</span> <span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">activation</span><span class="p">(</span><span class="n">query</span><span class="p">)</span> <span class="o">*</span> <span class="n">num</span> <span class="o">/</span> <span class="n">den</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>Output layer</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">221</span> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">output</span><span class="p">(</span><span class="n">y</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-28'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-28'>#</a>
|
|
</div>
|
|
<p>Test local mask</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">224</span><span class="k">def</span> <span class="nf">_test_local_mask</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>
|
|
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">228</span> <span class="kn">from</span> <span class="nn">labml.logger</span> <span class="kn">import</span> <span class="n">inspect</span>
|
|
<span class="lineno">229</span> <span class="n">inspect</span><span class="p">(</span><span class="n">AFTLocal</span><span class="o">.</span><span class="n">create_local_mask</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</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>
|
|
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">233</span><span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
|
|
<span class="lineno">234</span> <span class="n">_test_local_mask</span><span class="p">()</span></pre></div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
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<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js?config=TeX-AMS_HTML">
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