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<a class="parent" href="index.html">optimizers</a>
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<h1>AdaBelief Optimizer</h1>
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<p>This is based from AdaBelief
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<a href="https://github.com/juntang-zhuang/Adabelief-Optimizer">official implementation</a>
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of the paper
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<a href="https://arxiv.org/abs/2010.07468">AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients</a>.</p>
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<p>This is implemented in <a href="https://pytorch.org">PyTorch</a> as an extension to <a href="radam.html">RAdam</a>.</p>
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<p>The main difference between Adam optimizer and AdaBelief is that,
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how it calculates the adaptive learning rate;
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instead of dividing by the exponential moving average of square of the gradients,
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AdaBelief divides by the exponential mean of variance.</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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\color{cyan}{s_t} &\color{cyan}{\leftarrow} \color{cyan}{\beta_2 s_{t-1} + (1 - \beta_2) \cdot (g_t - m_t)^2} \\
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\hat{m}_t &\leftarrow \frac{m_t}{1-\beta_1^t} \\
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\color{cyan}{\hat{s}_t} &\color{cyan}{\leftarrow} \frac{\color{cyan}{s_t} + \color{red}{\epsilon}}{\color{cyan}{1-\beta_2^t}} \\
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\theta_t &\leftarrow \theta_{t-1} - \alpha \cdot \frac{\hat{m}_t}{\sqrt{\color{cyan}{\hat{s}_t}} + \epsilon}
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\end{align}</script>
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</p>
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<p>🤔 The paper calculates variance as $(g_t - m_t)^2$,
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but I feel it should use the bias corrected momentum
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$(g_t - \color{orange}{\hat{m}_t})^2$.
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I guess this doesn’t affect things much because
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bias correction is $\approx 1$ after the initial training steps.</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">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>
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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_nn.optimizers</span> <span class="kn">import</span> <span class="n">WeightDecay</span>
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<span class="lineno">42</span><span class="kn">from</span> <span class="nn">labml_nn.optimizers.radam</span> <span class="kn">import</span> <span class="n">RAdam</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>AdaBelief Optimizer</h2>
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<p>This class extends from RAdam optimizer defined in <a href="radam.html"><code>radam.py</code></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">45</span><span class="k">class</span> <span class="nc">AdaBelief</span><span class="p">(</span><span class="n">RAdam</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>‘optimized_update’ 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>amsgrad</code> is a flag indicating whether to use AMSGrad or fallback to plain Adam</li>
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<li><code>degenerate_to_sgd</code> whether to use sgd when the rectification term $r_t is intractable</li>
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<li>‘rectify’ is whether to use RAdam update</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>AdaBelief</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">52</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> <span class="n">lr</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">,</span> <span class="n">betas</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="o">=</span><span class="mf">1e-16</span><span class="p">,</span>
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<span class="lineno">53</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> <span class="n">amsgrad</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
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<span class="lineno">54</span> <span class="n">degenerate_to_sgd</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
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<span class="lineno">55</span> <span class="n">rectify</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">defaults</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>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">73</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">74</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">lr</span><span class="p">,</span> <span class="n">betas</span><span class="p">,</span> <span class="n">eps</span><span class="p">,</span> <span class="n">weight_decay</span><span class="p">,</span> <span class="n">amsgrad</span><span class="p">,</span> <span class="n">degenerate_to_sgd</span><span class="p">,</span> <span class="n">defaults</span><span class="p">)</span>
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<span class="lineno">75</span> <span class="bp">self</span><span class="o">.</span><span class="n">rectify</span> <span class="o">=</span> <span class="n">rectify</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">77</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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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">85</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 gradient values</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">87</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 variance</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">89</span> <span class="n">state</span><span class="p">[</span><span class="s1">'exp_avg_var'</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'>
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<div class='section-link'>
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<a href='#section-8'>#</a>
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</div>
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<p>If <code>amsgrad</code> flag is <code>True</code> for this parameter group, we maintain the maximum of
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exponential moving average of variance</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="k">if</span> <span class="n">group</span><span class="p">[</span><span class="s1">'amsgrad'</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>Maintains max of all exp. moving avg. of sq. grad. values</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">'max_exp_avg_var'</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-10'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-10'>#</a>
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</div>
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<h3>Calculate $m_t$ and $s_t$ or $\max(s_1, s_2, …, s_{t-1}, s_t)$</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">97</span> <span class="k">def</span> <span class="nf">get_ms</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-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>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">107</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-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>Get $m_{t-1}$ and $s_{t-1}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">110</span> <span class="n">m</span><span class="p">,</span> <span class="n">s</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_var'</span><span class="p">]</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-13'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-13'>#</a>
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</div>
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<p>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">114</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-14'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-14'>#</a>
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</div>
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<p>Difference between gradient and momentum</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">grad_residual</span> <span class="o">=</span> <span class="n">grad</span> <span class="o">-</span> <span class="n">m</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-15'>
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<div class='docs'>
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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>In-place calculation of $s_t$
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<script type="math/tex; mode=display">s_t \leftarrow \beta_2 s_{t-1} + (1 - \beta_2) \cdot (g_t - m_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">s</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_residual</span><span class="p">,</span> <span class="n">grad_residual</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></pre></div>
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</div>
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</div>
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<div class='section' id='section-16'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-16'>#</a>
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</div>
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<p>If this parameter group is using <code>amsgrad</code></p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">122</span> <span class="k">if</span> <span class="n">group</span><span class="p">[</span><span class="s1">'amsgrad'</span><span class="p">]:</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-17'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-17'>#</a>
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</div>
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<p>Get $\max(s_1, s_2, …, s_{t-1})$.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">124</span> <span class="n">s_max</span> <span class="o">=</span> <span class="n">state</span><span class="p">[</span><span class="s1">'max_exp_avg_var'</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-18'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-18'>#</a>
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</div>
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<p>Calculate $\max(s_1, s_2, …, s_{t-1}, s_t)$.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">126</span> <span class="n">torch</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">s_max</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">out</span><span class="o">=</span><span class="n">s_max</span><span class="p">)</span>
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<span class="lineno">127</span>
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<span class="lineno">128</span> <span class="k">return</span> <span class="n">m</span><span class="p">,</span> <span class="n">s_max</span>
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<span class="lineno">129</span> <span class="k">else</span><span class="p">:</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-19'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-19'>#</a>
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</div>
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<p>$m_t$ and $s_t$ otherwise</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">131</span> <span class="k">return</span> <span class="n">m</span><span class="p">,</span> <span class="n">s</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-20'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-20'>#</a>
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</div>
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<h3>Take an update step for a given parameter tensor</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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<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">133</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>
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</div>
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</div>
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|
<div class='section' id='section-21'>
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|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-21'>#</a>
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</div>
|
|
<p>Calculate weight decay</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">144</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>
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</div>
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</div>
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<div class='section' id='section-22'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-22'>#</a>
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</div>
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|
<p>Get $m_t$ and $v_t$</p>
|
|
</div>
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<div class='code'>
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|
<div class="highlight"><pre><span class="lineno">147</span> <span class="n">m</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_ms</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>
|
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</div>
|
|
</div>
|
|
<div class='section' id='section-23'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-23'>#</a>
|
|
</div>
|
|
<p>Increment $t$ the number of optimizer steps</p>
|
|
</div>
|
|
<div class='code'>
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|
<div class="highlight"><pre><span class="lineno">150</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>
|
|
<span class="lineno">151</span>
|
|
<span class="lineno">152</span> <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">rectify</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>Perform <em>Adam</em> update, defined in <a href="adam.html"><code>adam.py</code></a>, with
|
|
$\color{cyan}{s_t} + \color{red}{\epsilon}$ in place of $v_t$.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">155</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">s</span> <span class="o">+</span> <span class="n">group</span><span class="p">[</span><span class="s1">'eps'</span><span class="p">])</span>
|
|
<span class="lineno">156</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>Perform <em>Rectified Adam</em> update defined in <a href="radam.html"><code>radam.py</code></a>, with
|
|
$\color{cyan}{s_t} + \color{red}{\epsilon}$ in place of $v_t$.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">159</span> <span class="bp">self</span><span class="o">.</span><span class="n">r_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">s</span> <span class="o">+</span> <span class="n">group</span><span class="p">[</span><span class="s1">'eps'</span><span class="p">])</span></pre></div>
|
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</div>
|
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</div>
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</div>
|
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</div>
|
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<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js?config=TeX-AMS_HTML">
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</script>
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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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// we use CSS to left justify single line equations in code cells.
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