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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 Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation <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). This is a relative scheme tested on autoregressive tasks, and the bias is higher for closeby tokens and lower for far-away tokens. 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 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathnormal">i</span></span></span></span>-th token,</p>
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<span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.40006em;vertical-align:-2.45003em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95003em;"><span style="top:-4.95003em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.45003em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95003em;"><span style="top:-4.95003em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord text"><span class="mord">softmax</span></span><span class="mord"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mord mathbf">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">K</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">⊤</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="delimsizing size1">[</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mord"><span class="delimsizing size1">]</span></span><span class="mord"><span class="delimsizing size3">)</span></span></span></span><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord text"><span class="mord">softmax</span></span><span class="mord"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mord mathbf">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">K</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">⊤</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="delimsizing size1">[</span></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="delimsizing size1">]</span></span><span class="mord"><span class="delimsizing size3">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.45003em;"><span></span></span></span></span></span></span></span></span></span></span></span><p>where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathbf">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.849108em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span> is the query of the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathnormal">i</span></span></span></span>-th token, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span> are the keys up to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathnormal">i</span></span></span></span>, and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">d</span></span></span></span> the number of features per head. Note that the above equality halts because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">softmax</span></span></span></span></span> is invariant to translations (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 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">m</span></span></span></span> for each head</h2>
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<ul><li><code class="highlight"><span></span><span class="n">n_heads</span></code>
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is the number of heads in the attention layer <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span></span></span></span></li></ul>
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<p>The slope for first head is</p>
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<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0897em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0897em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9667142857142856em;"><span style="top:-2.9667142857142856em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5357142857142856em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mopen mtight">(</span><span class="mop mtight"><span class="mop mtight"><span class="mtight">l</span><span class="mtight">o</span><span class="mtight" style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3447999999999998em;margin-right:0.1em;"><span class="pstrut" style="height:2.64444em;"></span><span class="mord mtight">2</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29964em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.24533333333333335em;"></span><span class="mord mathnormal mtight">n</span><span class="mclose mtight">)</span><span class="mbin mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></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 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">8</span></span></span></span> the slopes are <span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<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>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-2'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-2'>#</a>
|
||
</div>
|
||
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0897em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0897em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9667142857142856em;"><span style="top:-2.9667142857142856em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5357142857142856em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mopen mtight">(</span><span class="mop mtight"><span class="mop mtight"><span class="mtight">l</span><span class="mtight">o</span><span class="mtight" style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3447999999999998em;margin-right:0.1em;"><span class="pstrut" style="height:2.64444em;"></span><span class="mord mtight">2</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29964em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.24533333333333335em;"></span><span class="mord mathnormal mtight">n</span><span class="mclose mtight">)</span><span class="mbin mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<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>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-3'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-3'>#</a>
|
||
</div>
|
||
<p>The geometric sequence </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<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>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-4'>
|
||
<div class='docs doc-strings'>
|
||
<div class='section-link'>
|
||
<a href='#section-4'>#</a>
|
||
</div>
|
||
<h2>Calculate the attention biases matrix</h2>
|
||
<ul><li><code class="highlight"><span></span><span class="n">n_heads</span></code>
|
||
is the number of heads in the attention layer </li>
|
||
<li><code class="highlight"><span></span><span class="n">max_len</span></code>
|
||
is the maximum sequence length</li></ul>
|
||
<p>This returns a matrix of shape <code class="highlight"><span></span><span class="p">[</span><span class="n">n_heads</span><span class="p">,</span> <span class="n">max_len</span><span class="p">]</span></code>
|
||
with attention biases.</p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<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>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-5'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-5'>#</a>
|
||
</div>
|
||
<p>Get slopes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">m</span></span></span></span> for each head </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<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>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-6'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-6'>#</a>
|
||
</div>
|
||
<p>Calculate distances <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mclose">]</span></span></span></span> </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<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>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-7'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-7'>#</a>
|
||
</div>
|
||
<p>Multiply them pair-wise to get the bias matrix </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<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>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-8'>
|
||
<div class='docs doc-strings'>
|
||
<div class='section-link'>
|
||
<a href='#section-8'>#</a>
|
||
</div>
|
||
<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 class="highlight"><span></span><span class="n">get_scores</span></code>
|
||
method.</p>
|
||
|
||
</div>
|
||
<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>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-9'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-9'>#</a>
|
||
</div>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<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>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-10'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-10'>#</a>
|
||
</div>
|
||
<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 class="highlight"><span></span><span class="p">[</span><span class="n">query_seq_len</span><span class="p">,</span> <span class="n">key_seq_len</span><span class="p">,</span> <span class="n">batch_size</span><span class="p">,</span> <span class="n">head</span><span class="p">]</span></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>
|
||
</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><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.093548em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathbf">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">K</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">⊤</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.44445em;vertical-align:0em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mord"><span class="delimsizing size1">[</span></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="delimsizing size1">]</span></span></span></span></span></span></p>
|
||
<p>Note that we add biases for all keys (not just upto <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathnormal">i</span></span></span></span>). 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 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> Simple test function to see the slopes.</p>
|
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</div>
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<div class='code'>
|
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<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>
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</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>
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</div>
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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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<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> </p>
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</div>
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<div class='code'>
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<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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