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<a class="parent" href="index.html">optimizers</a>
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<h1>Adam Optimizer</h1>
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<p>This is a <a href="https://pytorch.org">PyTorch</a> implementation of popular optimizer <em>Adam</em> from paper
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<a href="https://arxiv.org/abs/1412.6980v9">Adam: A Method for Stochastic Optimization</a>.</p>
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<p><em>Adam</em> update is,</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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m_t &\leftarrow \beta_1 m_{t-1} + (1 - \beta_1) \cdot g_t \\
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v_t &\leftarrow \beta_2 v_{t-1} + (1 - \beta_2) \cdot g_t^2 \\
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\hat{m}_t &\leftarrow \frac{m_t}{1-\beta_1^t} \\
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\hat{v}_t &\leftarrow \frac{v_t}{1-\beta_2^t} \\
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\theta_t &\leftarrow \theta_{t-1} - \alpha \cdot \frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon}
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\end{align}</script>
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</p>
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<p>where $\alpha$, $\beta_1$, $\beta_2$ and $\epsilon$ are scalar hyper parameters.
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$m_t$ and $v_t$ are first and second order moments.
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$\hat{m}_t$ and $\hat{v}_t$ are biased corrected moments.
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$\epsilon$ is used as a fix for division by zero error, but also acts as a form of a hyper-parameter
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that acts against variance in gradients.</p>
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<p>Effective step taken assuming $\epsilon = 0$ is,
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<script type="math/tex; mode=display">\Delta t = \alpha \cdot \frac{\hat{m}_t}{\hat{v}_t}</script>
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This is bounded by,
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<script type="math/tex; mode=display">\vert \Delta t \vert \le \alpha \cdot \frac{1 - \beta_1}{\sqrt{1-\beta_2}}</script>
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when $1-\beta_1 \gt \sqrt{1-\beta_2}$
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and
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<script type="math/tex; mode=display">\vert \Delta t\vert \le \alpha</script>
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otherwise.
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And in most common scenarios,
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<script type="math/tex; mode=display">\vert \Delta t \vert \approx \alpha</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">40</span><span></span><span class="kn">import</span> <span class="nn">math</span>
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<span class="lineno">41</span><span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Dict</span><span class="p">,</span> <span class="n">Any</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Optional</span>
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<span class="lineno">42</span>
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<span class="lineno">43</span><span class="kn">import</span> <span class="nn">torch</span>
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<span class="lineno">44</span><span class="kn">from</span> <span class="nn">labml</span> <span class="kn">import</span> <span class="n">tracker</span>
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<span class="lineno">45</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">46</span>
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<span class="lineno">47</span><span class="kn">from</span> <span class="nn">labml_nn.optimizers</span> <span class="kn">import</span> <span class="n">GenericAdaptiveOptimizer</span><span class="p">,</span> <span class="n">WeightDecay</span></pre></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>Adam Optimizer</h2>
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<p>We extend the class <code>GenericAdaptiveOptimizer</code> defined in <a href="index.html"><code>__init__.py</code></a>
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to implement the Adam optimizer.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">50</span><span class="k">class</span> <span class="nc">Adam</span><span class="p">(</span><span class="n">GenericAdaptiveOptimizer</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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<h3>Initialize the optimizer</h3>
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<ul>
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<li><code>params</code> is the list of parameters</li>
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<li><code>lr</code> is the learning rate $\alpha$</li>
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<li><code>betas</code> is a tuple of ($\beta_1$, $\beta_2$)</li>
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<li><code>eps</code> is $\hat{\epsilon}$ or $\epsilon$ based on <code>optimized_update</code></li>
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<li><code>weight_decay</code> is an instance of class <code>WeightDecay</code> defined in <a href="index.html"><code>__init__.py</code></a></li>
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<li><code>optimized_update</code> is a flag whether to optimize the bias correction of the second moment
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by doing it after adding $\epsilon$</li>
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<li><code>defaults</code> is a dictionary of default for group values.
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This is useful when you want to extend the class <code>Adam</code>.</li>
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</ul>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">58</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">params</span><span class="p">,</span>
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<span class="lineno">59</span> <span class="n">lr</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1e-3</span><span class="p">,</span> <span class="n">betas</span><span class="p">:</span> <span class="n">Tuple</span><span class="p">[</span><span class="nb">float</span><span class="p">,</span> <span class="nb">float</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.999</span><span class="p">),</span> <span class="n">eps</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1e-16</span><span class="p">,</span>
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<span class="lineno">60</span> <span class="n">weight_decay</span><span class="p">:</span> <span class="n">WeightDecay</span> <span class="o">=</span> <span class="n">WeightDecay</span><span class="p">(),</span>
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<span class="lineno">61</span> <span class="n">optimized_update</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="kc">True</span><span class="p">,</span>
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<span class="lineno">62</span> <span class="n">defaults</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="n">Any</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-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 class='code'>
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<div class="highlight"><pre><span class="lineno">76</span> <span class="n">defaults</span> <span class="o">=</span> <span class="p">{}</span> <span class="k">if</span> <span class="n">defaults</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">defaults</span>
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<span class="lineno">77</span> <span class="n">defaults</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">weight_decay</span><span class="o">.</span><span class="n">defaults</span><span class="p">())</span>
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<span class="lineno">78</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">params</span><span class="p">,</span> <span class="n">defaults</span><span class="p">,</span> <span class="n">lr</span><span class="p">,</span> <span class="n">betas</span><span class="p">,</span> <span class="n">eps</span><span class="p">)</span>
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<span class="lineno">79</span>
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<span class="lineno">80</span> <span class="bp">self</span><span class="o">.</span><span class="n">weight_decay</span> <span class="o">=</span> <span class="n">weight_decay</span>
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<span class="lineno">81</span> <span class="bp">self</span><span class="o">.</span><span class="n">optimized_update</span> <span class="o">=</span> <span class="n">optimized_update</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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<h3>Initialize a parameter state</h3>
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<ul>
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<li><code>state</code> is the optimizer state of the parameter (tensor)</li>
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<li><code>group</code> stores optimizer attributes of the parameter group</li>
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<li><code>param</code> is the parameter tensor $\theta_{t-1}$</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">83</span> <span class="k">def</span> <span class="nf">init_state</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">],</span> <span class="n">group</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">],</span> <span class="n">param</span><span class="p">:</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</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>This is the number of optimizer steps taken on the parameter, $t$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">93</span> <span class="n">state</span><span class="p">[</span><span class="s1">'step'</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-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>Exponential moving average of gradients, $m_t$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">95</span> <span class="n">state</span><span class="p">[</span><span class="s1">'exp_avg'</span><span class="p">]</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">memory_format</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">preserve_format</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>Exponential moving average of squared gradient values, $v_t$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">97</span> <span class="n">state</span><span class="p">[</span><span class="s1">'exp_avg_sq'</span><span class="p">]</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">memory_format</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">preserve_format</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'>
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<a href='#section-8'>#</a>
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</div>
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<h3>Calculate $m_t$ and and $v_t$</h3>
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<ul>
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<li><code>state</code> is the optimizer state of the parameter (tensor)</li>
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<li><code>group</code> stores optimizer attributes of the parameter group</li>
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<li><code>grad</code> is the current gradient tensor $g_t$ for the parameter $\theta_{t-1}$</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">99</span> <span class="k">def</span> <span class="nf">get_mv</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="n">Any</span><span class="p">],</span> <span class="n">group</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="n">Any</span><span class="p">],</span> <span class="n">grad</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-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>Get $\beta_1$ and $\beta_2$</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="n">beta1</span><span class="p">,</span> <span class="n">beta2</span> <span class="o">=</span> <span class="n">group</span><span class="p">[</span><span class="s1">'betas'</span><span class="p">]</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-10'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-10'>#</a>
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</div>
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<p>Get $m_{t-1}$ and $v_{t-1}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">112</span> <span class="n">m</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">state</span><span class="p">[</span><span class="s1">'exp_avg'</span><span class="p">],</span> <span class="n">state</span><span class="p">[</span><span class="s1">'exp_avg_sq'</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>In-place calculation of $m_t$
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<script type="math/tex; mode=display">m_t \leftarrow \beta_1 m_{t-1} + (1 - \beta_1) \cdot g_t</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">116</span> <span class="n">m</span><span class="o">.</span><span class="n">mul_</span><span class="p">(</span><span class="n">beta1</span><span class="p">)</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="n">grad</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span> <span class="o">-</span> <span class="n">beta1</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>In-place calculation of $v_t$
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<script type="math/tex; mode=display">v_t \leftarrow \beta_2 v_{t-1} + (1 - \beta_2) \cdot g_t^2</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">119</span> <span class="n">v</span><span class="o">.</span><span class="n">mul_</span><span class="p">(</span><span class="n">beta2</span><span class="p">)</span><span class="o">.</span><span class="n">addcmul_</span><span class="p">(</span><span class="n">grad</span><span class="p">,</span> <span class="n">grad</span><span class="p">,</span> <span class="n">value</span><span class="o">=</span><span class="mi">1</span> <span class="o">-</span> <span class="n">beta2</span><span class="p">)</span>
|
|
<span class="lineno">120</span>
|
|
<span class="lineno">121</span> <span class="k">return</span> <span class="n">m</span><span class="p">,</span> <span class="n">v</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-13'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-13'>#</a>
|
|
</div>
|
|
<h3>Get learning-rate</h3>
|
|
<p>This returns the modified learning rate based on the state.
|
|
For <em>Adam</em> this is just the specified learning rate for the parameter group,
|
|
$\alpha$.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">123</span> <span class="k">def</span> <span class="nf">get_lr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">],</span> <span class="n">group</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</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>
|
|
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">131</span> <span class="k">return</span> <span class="n">group</span><span class="p">[</span><span class="s1">'lr'</span><span class="p">]</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-15'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-15'>#</a>
|
|
</div>
|
|
<h3>Do the <em>Adam</em> parameter update</h3>
|
|
<ul>
|
|
<li><code>state</code> is the optimizer state of the parameter (tensor)</li>
|
|
<li><code>group</code> stores optimizer attributes of the parameter group</li>
|
|
<li><code>param</code> is the parameter tensor $\theta_{t-1}$</li>
|
|
<li><code>m</code> and <code>v</code> are the uncorrected first and second moments $m_t$ and $v_t$.</li>
|
|
</ul>
|
|
<p>This computes the following</p>
|
|
<p>
|
|
<script type="math/tex; mode=display">\begin{align}
|
|
\theta_t &\leftarrow \theta_{t-1} - \alpha \cdot \frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon}
|
|
\end{align}</script>
|
|
</p>
|
|
<p>Since $\alpha$, $\beta_1$, $\beta_2$ and $\epsilon$ are scalars and others are tensors
|
|
we modify this calculation to optimize the computation.</p>
|
|
<p>
|
|
<script type="math/tex; mode=display">\begin{align}
|
|
\theta_t &\leftarrow \theta_{t-1} - \alpha \cdot \frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon} \\
|
|
\theta_t &\leftarrow \theta_{t-1} - \alpha \cdot
|
|
\frac{m_t / (1-\beta_1^t)}{\sqrt{v_t/(1-\beta_2^t)} + \epsilon} \\
|
|
\theta_t &\leftarrow \theta_{t-1} - \alpha \frac{\sqrt{1-\beta_2^t}}{1-\beta_1^t} \cdot
|
|
\frac{m_t}{\sqrt{v_t} + \hat{\epsilon}} \\
|
|
\end{align}</script>
|
|
</p>
|
|
<p>where
|
|
<script type="math/tex; mode=display">\hat{\epsilon} = (1-\beta_2^t) \epsilon</script>
|
|
is what we should specify as the hyper-parameter.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">133</span> <span class="k">def</span> <span class="nf">adam_update</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">],</span> <span class="n">group</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">],</span> <span class="n">param</span><span class="p">:</span> <span class="n">torch</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="lineno">134</span> <span class="n">m</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">v</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-16'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-16'>#</a>
|
|
</div>
|
|
<p>Get $\beta_1$ and $\beta_2$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">166</span> <span class="n">beta1</span><span class="p">,</span> <span class="n">beta2</span> <span class="o">=</span> <span class="n">group</span><span class="p">[</span><span class="s1">'betas'</span><span class="p">]</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-17'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-17'>#</a>
|
|
</div>
|
|
<p>Bias correction term for $\hat{m}_t$, $1 - \beta_1^t$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">168</span> <span class="n">bias_correction1</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">beta1</span> <span class="o">**</span> <span class="n">state</span><span class="p">[</span><span class="s1">'step'</span><span class="p">]</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-18'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-18'>#</a>
|
|
</div>
|
|
<p>Bias correction term for $\hat{v}_t$, $1 - \beta_2^t$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">170</span> <span class="n">bias_correction2</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">beta2</span> <span class="o">**</span> <span class="n">state</span><span class="p">[</span><span class="s1">'step'</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 learning rate</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">173</span> <span class="n">lr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_lr</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">group</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>Whether to optimize the computation</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">176</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">optimized_update</span><span class="p">:</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-21'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-21'>#</a>
|
|
</div>
|
|
<p>$\sqrt{v_t} + \hat{\epsilon}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">178</span> <span class="n">denominator</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="n">group</span><span class="p">[</span><span class="s1">'eps'</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>$\alpha \frac{\sqrt{1-\beta_2^t}}{1-\beta_1^t}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">180</span> <span class="n">step_size</span> <span class="o">=</span> <span class="n">lr</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">bias_correction2</span><span class="p">)</span> <span class="o">/</span> <span class="n">bias_correction1</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>$\theta_t \leftarrow \theta_{t-1} - \alpha \frac{\sqrt{1-\beta_2^t}}{1-\beta_1^t} \cdot
|
|
\frac{m_t}{\sqrt{v_t} + \hat{\epsilon}}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">183</span> <span class="n">param</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">addcdiv_</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">denominator</span><span class="p">,</span> <span class="n">value</span><span class="o">=-</span><span class="n">step_size</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>Computation without optimization</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">185</span> <span class="k">else</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>$\frac{\sqrt{v_t}}{\sqrt{1-\beta_2^t}} + \epsilon$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">187</span> <span class="n">denominator</span> <span class="o">=</span> <span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span> <span class="o">/</span> <span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">bias_correction2</span><span class="p">))</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="n">group</span><span class="p">[</span><span class="s1">'eps'</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>$\frac{\alpha}{1-\beta_1^t}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">189</span> <span class="n">step_size</span> <span class="o">=</span> <span class="n">lr</span> <span class="o">/</span> <span class="n">bias_correction1</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>$\theta_t \leftarrow \theta_{t-1} - \alpha \cdot
|
|
\frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">192</span> <span class="n">param</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">addcdiv_</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">denominator</span><span class="p">,</span> <span class="n">value</span><span class="o">=-</span><span class="n">step_size</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>
|
|
<h3>Take an update step for a given parameter tensor</h3>
|
|
<ul>
|
|
<li><code>state</code> is the optimizer state of the parameter (tensor)</li>
|
|
<li><code>group</code> stores optimizer attributes of the parameter group</li>
|
|
<li><code>grad</code> is the current gradient tensor $g_t$ for the parameter $\theta_{t-1}$</li>
|
|
<li><code>param</code> is the parameter tensor $\theta_{t-1}$</li>
|
|
</ul>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">194</span> <span class="k">def</span> <span class="nf">step_param</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">],</span> <span class="n">group</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">],</span> <span class="n">grad</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">param</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">):</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-29'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-29'>#</a>
|
|
</div>
|
|
<p>Calculate weight decay</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">205</span> <span class="n">grad</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">weight_decay</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">grad</span><span class="p">,</span> <span class="n">group</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-30'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-30'>#</a>
|
|
</div>
|
|
<p>Get $m_t$ and $v_t$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">208</span> <span class="n">m</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_mv</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">group</span><span class="p">,</span> <span class="n">grad</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-31'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-31'>#</a>
|
|
</div>
|
|
<p>Increment $t$ the number of optimizer steps</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">211</span> <span class="n">state</span><span class="p">[</span><span class="s1">'step'</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-32'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-32'>#</a>
|
|
</div>
|
|
<p>Perform <em>Adam</em> update</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">214</span> <span class="bp">self</span><span class="o">.</span><span class="n">adam_update</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">group</span><span class="p">,</span> <span class="n">param</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js?config=TeX-AMS_HTML">
|
|
</script>
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<!-- MathJax configuration -->
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<script type="text/x-mathjax-config">
|
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MathJax.Hub.Config({
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tex2jax: {
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inlineMath: [ ['$','$'] ],
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"HTML-CSS": { fonts: ["TeX"] }
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|
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modalContent.appendChild(modalImage)
|
|
|
|
var span = document.createElement('span')
|
|
span.classList.add('close')
|
|
span.textContent = 'x'
|
|
modal.appendChild(span)
|
|
|
|
img.onclick = function () {
|
|
console.log('clicked')
|
|
document.body.appendChild(modal)
|
|
modalImage.src = img.src
|
|
}
|
|
|
|
span.onclick = function () {
|
|
document.body.removeChild(modal)
|
|
}
|
|
}
|
|
|
|
handleImages()
|
|
</script>
|
|
</body>
|
|
</html> |