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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">alibi</a>
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<a href="https://github.com/labmlai/annotated_deep_learning_paper_implementations/tree/master/labml_nn/transformers/alibi/__init__.py">
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<a href='#section-0'>#</a>
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</div>
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<h1>Attention with Linear Biases (ALiBi)</h1>
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<p>This is an implementation of Attention with Linear Biases (ALiBi) from the paper
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Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation
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<a href="https://ofir.io/train_short_test_long.pdf">(pdf)</a>.</p>
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<p>This replaces positional encodings with biases added to attention scores (attention logits, before the softmax).
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This is a relative scheme tested on autoregressive tasks, and the bias is higher for closeby tokens
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and lower for far-away tokens.
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The biases decrease linearly in the log scale (because it’s before the softmax) and each head has a different slope.</p>
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<p>Here’s the attention formula for $i$-th token,</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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\mathbf{a}_i
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&= \text{softmax} \bigg( \mathbf{q}_i \mathbf{K}^\top + m \cdot \big[-(i-1), \dots, 1, 0 \big] \bigg) \\
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&= \text{softmax} \bigg( \mathbf{q}_i \mathbf{K}^\top + m \cdot \big[0, 1, \dots, (i - 1) \big] \bigg)
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\end{align}</script>
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</p>
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<p>where $\mathbf{q}_i \in \mathbb{R}^d$ is the query of the $i$-th token, $K \in \mathbb{R}^{i \times d}$ are the keys
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up to $i$, and $d$ the number of features per head.
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Note that the above equality halts because $\text{softmax}$ is invariant to translations
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(you can add any constant to all elements without changing the result).</p>
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<p>Here is <a href="experiment.html">the training code</a> for a ALiBi model.</p>
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<p><a href="https://app.labml.ai/run/e87bec2a074911ec82cdd1759f10c925"><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">36</span><span></span><span class="kn">import</span> <span class="nn">math</span>
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<span class="lineno">37</span>
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<span class="lineno">38</span><span class="kn">import</span> <span class="nn">torch</span>
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<span class="lineno">39</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">40</span>
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<span class="lineno">41</span><span class="kn">from</span> <span class="nn">labml.logger</span> <span class="kn">import</span> <span class="n">inspect</span>
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<span class="lineno">42</span><span class="kn">from</span> <span class="nn">labml_nn.transformers.mha</span> <span class="kn">import</span> <span class="n">MultiHeadAttention</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>Get head-specific slope $m$ for each head</h2>
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<ul>
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<li><code>n_heads</code> is the number of heads in the attention layer $n$</li>
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</ul>
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<p>The slope for first head is</p>
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<p>
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<script type="math/tex; mode=display">2^{-2^{-(\log_2 n) - 3}}</script>
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</p>
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<p>The slopes for the rest of the heads are in a geometric series with a ratio same as above.</p>
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<p>For instance when the number of heads is $8$ the slopes are
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<script type="math/tex; mode=display">\frac{1}{2^1}, \frac{1}{2^2}, \dots, \frac{1}{2^8}</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">45</span><span class="k">def</span> <span class="nf">get_slopes</span><span class="p">(</span><span class="n">n_heads</span><span class="p">:</span> <span class="nb">int</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'>
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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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<p>
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<script type="math/tex; mode=display">2^{-2^{-(\log_2 n) - 3}}</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">62</span> <span class="n">s</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span> <span class="o">**</span> <span class="p">(</span><span class="o">-</span><span class="mi">2</span> <span class="o">**</span> <span class="o">-</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log2</span><span class="p">(</span><span class="n">n_heads</span><span class="p">)</span> <span class="o">-</span> <span class="mi">3</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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<p>The geometric sequence</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">64</span> <span class="k">return</span> <span class="p">[</span><span class="n">s</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span> <span class="o">**</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_heads</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 doc-strings'>
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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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<h2>Calculate the attention biases matrix</h2>
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<ul>
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<li><code>n_heads</code> is the number of heads in the attention layer</li>
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<li><code>max_len</code> is the maximum sequence length</li>
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</ul>
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<p>This returns a matrix of shape <code>[n_heads, max_len]</code> with attention biases.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">67</span><span class="k">def</span> <span class="nf">get_biases</span><span class="p">(</span><span class="n">n_heads</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">max_len</span><span class="p">:</span> <span class="nb">int</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-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>Get slopes $m$ 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">78</span> <span class="n">slopes</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">get_slopes</span><span class="p">(</span><span class="n">n_heads</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>Calculate distances $[0, 1, \dots, N]$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">80</span> <span class="n">distance</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">max_len</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">float</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>Multiply them pair-wise to get the bias matrix</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">82</span> <span class="k">return</span> <span class="n">distance</span><span class="p">[:,</span> <span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">slopes</span><span class="p">[</span><span class="kc">None</span><span class="p">,</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 doc-strings'>
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<div class='section-link'>
|
||||
<a href='#section-8'>#</a>
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</div>
|
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<h2>Attention with Linear Biases (ALiBi)</h2>
|
||||
<p>We override <a href="mha.html">Multi-Head Attention</a> module so we only need to
|
||||
write the <code>get_scores</code> method.</p>
|
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</div>
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<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">85</span><span class="k">class</span> <span class="nc">AlibiMultiHeadAttention</span><span class="p">(</span><span class="n">MultiHeadAttention</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'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-9'>#</a>
|
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</div>
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|
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</div>
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<div class='code'>
|
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<div class="highlight"><pre><span class="lineno">93</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">heads</span><span class="p">:</span> <span class="nb">int</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">dropout_prob</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span> <span class="n">max_len</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">5_000</span><span class="p">):</span>
|
||||
<span class="lineno">94</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">heads</span><span class="p">,</span> <span class="n">d_model</span><span class="p">,</span> <span class="n">dropout_prob</span><span class="p">)</span></pre></div>
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</div>
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</div>
|
||||
<div class='section' id='section-10'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-10'>#</a>
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</div>
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<p>Pre-calculate the biases</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">97</span> <span class="bp">self</span><span class="o">.</span><span class="n">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">get_biases</span><span class="p">(</span><span class="n">heads</span><span class="p">,</span> <span class="n">max_len</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>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-11'>
|
||||
<div class='docs doc-strings'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-11'>#</a>
|
||||
</div>
|
||||
<h3>Calculate attention scores and add attention biases</h3>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">99</span> <span class="k">def</span> <span class="nf">get_scores</span><span class="p">(</span><span class="bp">self</span><span class="p">,</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="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></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-12'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-12'>#</a>
|
||||
</div>
|
||||
<p>Calculate the standard attention score.
|
||||
It has shape <code>[query_seq_len, key_seq_len, batch_size, head]</code></p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">106</span> <span class="n">scores</span> <span class="o">=</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="n">get_scores</span><span class="p">(</span><span class="n">query</span><span class="p">,</span> <span class="n">key</span><span class="p">)</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-13'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-13'>#</a>
|
||||
</div>
|
||||
<p>Number of keys</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">109</span> <span class="n">key_seq_len</span> <span class="o">=</span> <span class="n">scores</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>
|
||||
</div>
|
||||
<div class='section' id='section-14'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-14'>#</a>
|
||||
</div>
|
||||
<p>Add the biases to scores.</p>
|
||||
<p>
|
||||
<script type="math/tex; mode=display">\mathbf{q}_i \mathbf{K}^\top + m \cdot \big[0, 1, \dots, (i - 1) \big]</script>
|
||||
</p>
|
||||
<p>Note that we add biases for all keys (not just upto $i$). We can do this since
|
||||
those extra entries will get removed because of the masking later.</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">116</span> <span class="k">return</span> <span class="n">scores</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">bias</span><span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="p">:</span><span class="n">key_seq_len</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</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 doc-strings'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-15'>#</a>
|
||||
</div>
|
||||
<p>Simple test function to see the slopes.</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">119</span><span class="k">def</span> <span class="nf">_test_slopes</span><span class="p">():</span></pre></div>
|
||||
</div>
|
||||
</div>
|
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<div class='section' id='section-16'>
|
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<div class='docs'>
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<a href='#section-16'>#</a>
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<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">123</span> <span class="n">inspect</span><span class="p">(</span><span class="n">get_slopes</span><span class="p">(</span><span class="mi">8</span><span class="p">))</span>
|
||||
<span class="lineno">124</span> <span class="n">inspect</span><span class="p">(</span><span class="n">get_slopes</span><span class="p">(</span><span class="mi">16</span><span class="p">))</span></pre></div>
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<a href='#section-17'>#</a>
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<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">128</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">129</span> <span class="n">_test_slopes</span><span class="p">()</span></pre></div>
|
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