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<h1>Rectified Adam (RAdam) optimizer</h1>
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<p>This implementation is based on
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<a href="https://github.com/LiyuanLucasLiu/RAdam">the official implementation</a>
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of the paper
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<a href="https://arxiv.org/abs/1908.03265">On the Variance of the Adaptive Learning Rate and Beyond</a>.</p>
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<p>We have implemented it in <a href="https://pytorch.org">PyTorch</a>
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as an extension to <a href="amsgrad.html">our AMSGrad implementation</a>
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thus requiring only the modifications to be implemented.</p>
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<p>Adam optimizer sometimes converges to a bad local optima during the initial stages of the training;
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especially when training transformers.
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Researches use warmups to counter this; for the the initial training steps (warm-up stage)
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they use a low learning rate.
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This paper identifies the problem to be the high variance of adaptive learning rate
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during initial stages of training, and counters it using a new rectification term to
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reduce variance.</p>
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<p>The paper also evaluates two variance reduction mechanisms:
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* <strong>Adam-2k</strong>: Only compute the adaptive learning rate ($v_t$ in <a href="adam.html">Adam</a>) during the first 2k steps,
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without changing parameters or calculating momentum ($m_t$).
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* <strong>Adam-eps</strong>: Adam with large $\epsilon \approx 10^{-4}$.</p>
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<h2>Rectified Adam</h2>
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<p>Let $\sigma(g_1, …, g_t)$ and $\psi(g_1, …, g_t)$ be the functions to calculate
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momentum and adaptive learning rate.
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For Adam, they are
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<script type="math/tex; mode=display">\begin{align}
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\sigma(g_1, ..., g_t) &= \frac{(1 - \beta_1)\sum_{i=1}^t \beta_1^{t-i} g_i}{1 - \beta_1^t} \\
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\psi(g_1, ..., g_t) &= \sqrt \frac{1 - \beta_2^t}{(1 - \beta_2)\sum_{i=1}^t \beta_2^{t-i} g_i^2}
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\end{align}</script>
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</p>
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<h3>Exponential moving average as simple moving average</h3>
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<p>The distribution of exponential moving average can be approximated as a simple moving average.
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<script type="math/tex; mode=display">\begin{align}
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p\Bigg(\frac{(1-\beta_2) \sum_{i=1}^t \beta_2^{t-i} g_i^2}{1 - \beta_2^t} \Bigg) \approx
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p\Bigg(\frac{\sum_{i=1}^{f(t,\beta_2)} g_{t+1-i}^2}{f(t,\beta_2)} \Bigg)
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\end{align}</script>
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Here we are taking the simple moving average of the last $f(t,\beta_2)$ gradients.
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$f(t,\beta_2)$ satisfies the following,
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<script type="math/tex; mode=display">\begin{align}
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\frac{(1-\beta_2) \sum_{i=1}^t \beta_2^{t-i} \cdot i}{1 - \beta_2^t} =
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\frac{\sum_{i=1}^{f(t,\beta_2)} (t+1-i)}{f(t,\beta_2)}
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\end{align}</script>
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which gives,
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<script type="math/tex; mode=display">f(t,\beta_2) = \frac{2}{1-\beta_2} - 1 - \frac{2 t \beta_2^t}{1 - \beta_2^t}</script>
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</p>
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<h3>Scaled inverse chi-squared</h3>
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<p>From above we have
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<script type="math/tex; mode=display">
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p\Big( \psi^2(g_1, ..., g_t) \Big) \approx
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p\Bigg(\frac{\sum_{i=1}^{f(t,\beta_2)} g_{t+1-i}^2}{f(t,\beta_2)} \Bigg)
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</script>
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where $g_i \sim \mathcal{N}(0, \sigma^2)$.
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Note that $sigma$ here is the standard deviation and different from $\sigma(.)$ for momentum.</p>
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<p><a href="https://en.wikipedia.org/wiki/Scaled_inverse_chi-squared_distribution">Scaled inverse chi-squared</a>
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is the distribution of squared inverse of mean of $p$ normal distributions.
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<script type="math/tex; mode=display">
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p\Bigg(\frac{\sum_{i=1}^{f(t,\beta_2)} g_{t+1-i}^2}{f(t,\beta_2)} \Bigg)
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\sim
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\text{Scale-inv} \mathcal{X}^2(\rho,\frac{1}{\sigma^2})
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</script>
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where $\rho = f(t,\beta_2)$.</p>
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<h3>Rectification</h3>
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<p>They prove that variance of $\psi(.)$ decreases with $\rho$ when
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$\psi^2(.) \sim \text{Scale-inv} \mathcal{X}^2(\rho,\frac{1}{\sigma^2})$.</p>
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<p>Therefore the variance is minimized at maximal $\rho$ which is
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$\rho_{\infty} = \frac{2}{1-\beta_2} - 1$. Let the minimum variance be $C_{\text{var}}$</p>
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<p>In order to ensure that the adaptive learning
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rate $\psi(.)$ has consistent variance, we rectify the variance with $r$
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<script type="math/tex; mode=display">\begin{align}
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r = \sqrt{\frac{C_{\text{var}}}{Var\big[\psi(.)\big]}}
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\end{align}</script>
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</p>
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<h3>Approximating $Var[\psi(.)]$</h3>
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<p>They estimate $Var[\psi(.)] \approx \frac{Var[\psi^2(.)]}{4 \mathbb{E}[\psi^2(.)}$
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based on first order expansion of $\sqrt{\psi^2(.)}$
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🤪 I didn’t get how it was derived.</p>
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<p>From $\text{Scale-inv} \mathcal{X}^2$ distribution we have,
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<script type="math/tex; mode=display">\begin{align}
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\mathbb{E}\big[\psi^2(.)\big] &= \frac{\rho / \sigma^2}{\rho-2} \\
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Var\big[\psi^2(.)\big] &= \frac{2 \rho / \sigma^4}{(\rho-2)^2 (\rho - 2)}
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\end{align}</script>
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which gives,
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<script type="math/tex; mode=display">
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Var[\psi(.)] \approx \frac{\rho}{2(\rho-2)(\rho-4)\sigma^2}
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</script>
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</p>
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<h3>Rectification term</h3>
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<p>We have
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<script type="math/tex; mode=display">\begin{align}
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r &= \sqrt{\frac{C_{\text{var}}}{Var\big[\psi(.)\big]}} \\
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Var[\psi(.)] &\approx \frac{\rho}{2(\rho-2)(\rho-4)\sigma^2}
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\end{align}</script>
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</p>
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<p>where $C_{\text{var}}$ is $Var\big[\psi(.)\big]$ for $\rho_\infty$.
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Lt $\rho$ and step $t$ be $\rho_t$, and $r_t$ be the rectification term
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at step $t$.</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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C_{\text{var}} &\approx \frac{\rho_\infty}{2(\rho_\infty-2)(\rho_\infty-4)\sigma^2} \\
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Var[\psi(g_1,...,g_t)] &\approx \frac{\rho_t}{2(\rho_t-2)(\rho_t-4)\sigma^2}
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\end{align}</script>
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</p>
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<p>This gives,
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<script type="math/tex; mode=display">\begin{align}
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r_t &= \sqrt{\frac{(\rho_t-2)(\rho_t-4)\rho_\infty}{(\rho_\infty-2)(\rho_\infty-4)\rho_t}}
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\end{align}</script>
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</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">129</span><span></span><span class="kn">import</span> <span class="nn">math</span>
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<span class="lineno">130</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">Optional</span>
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<span class="lineno">131</span>
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<span class="lineno">132</span><span class="kn">import</span> <span class="nn">torch</span>
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<span class="lineno">133</span>
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<span class="lineno">134</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">135</span><span class="kn">from</span> <span class="nn">labml_nn.optimizers.amsgrad</span> <span class="kn">import</span> <span class="n">AMSGrad</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>Rectified Adam Optimizer</h2>
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<p>This class extends from AMSAdam optimizer defined in <a href="amsadam.html"><code>amsadam.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">138</span><span class="k">class</span> <span class="nc">RAdam</span><span class="p">(</span><span class="n">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-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><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>RAdam</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">145</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-8</span><span class="p">,</span>
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<span class="lineno">146</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">147</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">148</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">149</span> <span class="n">degenerated_to_sgd</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">165</span> <span class="bp">self</span><span class="o">.</span><span class="n">degenerated_to_sgd</span> <span class="o">=</span> <span class="n">degenerated_to_sgd</span>
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<span class="lineno">166</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">optimized_update</span><span class="p">,</span> <span class="n">amsgrad</span><span class="p">,</span> <span class="n">defaults</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-4'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-4'>#</a>
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</div>
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<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">168</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-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>Calculate weight decay</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">179</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-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>Get $m_t$ and $v_t$; i.e. $\sigma(.)$ and $\psi(.)$ without bias correction</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">182</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>
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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>Calculate $t$ the number of optimizer 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">185</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>
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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>Perform <em>RAdam</em> update</p>
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</div>
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<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">188</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">v</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-9'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-9'>#</a>
|
|
</div>
|
|
<h3>Calculate rectification term $r_t$</h3>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">190</span> <span class="nd">@staticmethod</span>
|
|
<span class="lineno">191</span> <span class="k">def</span> <span class="nf">calc_rectification_term</span><span class="p">(</span><span class="n">beta2</span><span class="p">:</span> <span class="nb">float</span><span class="p">,</span> <span class="n">step</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="n">Optional</span><span class="p">[</span><span class="nb">float</span><span class="p">]:</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-10'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
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<a href='#section-10'>#</a>
|
|
</div>
|
|
<p>$\beta_2^t$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">197</span> <span class="n">beta2_t</span> <span class="o">=</span> <span class="n">beta2</span> <span class="o">**</span> <span class="n">step</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-11'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-11'>#</a>
|
|
</div>
|
|
<p>
|
|
<script type="math/tex; mode=display">\rho_\infty = \frac{2}{1 - \beta_2} - 1</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">199</span> <span class="n">rho_inf</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">beta2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span></pre></div>
|
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</div>
|
|
</div>
|
|
<div class='section' id='section-12'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-12'>#</a>
|
|
</div>
|
|
<p>
|
|
<script type="math/tex; mode=display">\rho_t = \frac{2}{1-\beta_2} - 1 - \frac{2 t \beta_2^t}{1-\beta_2^t}</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">201</span> <span class="n">rho</span> <span class="o">=</span> <span class="n">rho_inf</span> <span class="o">-</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">step</span> <span class="o">*</span> <span class="n">beta2_t</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">beta2_t</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>$r_t$ is tractable when $\rho_t >= 4$.
|
|
We are being a little more conservative since it’s an approximated value</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">205</span> <span class="k">if</span> <span class="n">rho</span> <span class="o">>=</span> <span class="mi">5</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>
|
|
<script type="math/tex; mode=display">r_t = \sqrt{\frac{(\rho_t-2)(\rho_t-4)\rho_\infty}{(\rho_\infty-2)(\rho_\infty-4)\rho_t}}</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">207</span> <span class="n">r2</span> <span class="o">=</span> <span class="p">(</span><span class="n">rho</span> <span class="o">-</span> <span class="mi">4</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">rho_inf</span> <span class="o">-</span> <span class="mi">4</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">rho</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="n">rho</span> <span class="o">*</span> <span class="n">rho_inf</span> <span class="o">/</span> <span class="p">(</span><span class="n">rho_inf</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span>
|
|
<span class="lineno">208</span> <span class="k">return</span> <span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">r2</span><span class="p">)</span>
|
|
<span class="lineno">209</span> <span class="k">else</span><span class="p">:</span>
|
|
<span class="lineno">210</span> <span class="k">return</span> <span class="kc">None</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>RAdam</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 $v_t$;
|
|
i.e. $\sigma(.)$ and $\psi(.)$ without bias correction</li>
|
|
</ul>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">212</span> <span class="k">def</span> <span class="nf">r_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">213</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">225</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">227</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">229</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>
|
|
<span class="lineno">230</span>
|
|
<span class="lineno">231</span> <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">calc_rectification_term</span><span class="p">(</span><span class="n">beta2</span><span class="p">,</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">234</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>If $r_t$ is intractable</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">237</span> <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</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>Whether to optimize the computation by combining scalar computations</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">239</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-22'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-22'>#</a>
|
|
</div>
|
|
<p>Denominator $\sqrt{v_t} + \hat{\epsilon}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">241</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-23'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-23'>#</a>
|
|
</div>
|
|
<p>Step size $\alpha \sqrt{r_t} * \frac{\sqrt{1-\beta_2^t}}{1-\beta_1^t}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">243</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">r</span> <span class="o">/</span> <span class="n">bias_correction1</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>Update parameters $\theta_t \leftarrow \theta_{t-1} - \alpha \sqrt{r_t} \frac{\sqrt{1-\beta_2^t}}{1-\beta_1^t} \cdot
|
|
\frac{m_t}{\sqrt{v_t} + \hat{\epsilon}}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">246</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-25'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-25'>#</a>
|
|
</div>
|
|
<p>Computation without optimization</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">248</span> <span class="k">else</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>Denominator $\frac{\sqrt{v_t}}{\sqrt{1-\beta_2^t}} + \epsilon$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">250</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-27'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-27'>#</a>
|
|
</div>
|
|
<p>Step size $\frac{\alpha \sqrt{r_t}}{1-\beta_1^t}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">252</span> <span class="n">step_size</span> <span class="o">=</span> <span class="n">lr</span> <span class="o">*</span> <span class="n">r</span> <span class="o">/</span> <span class="n">bias_correction1</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-28'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-28'>#</a>
|
|
</div>
|
|
<p>Update parameters $\theta_t \leftarrow \theta_{t-1} - \alpha \sqrt{r_t} \cdot
|
|
\frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">255</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-29'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-29'>#</a>
|
|
</div>
|
|
<p>If $r_t$ is intractable do a SGD with momentum</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">258</span> <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">degenerated_to_sgd</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>Step size $\frac{\alpha}{1-\beta_1^t}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">260</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-31'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-31'>#</a>
|
|
</div>
|
|
<p>Update parameters
|
|
$\theta_t \leftarrow \theta_{t-1} - \alpha \cdot \hat{m}_t$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">263</span> <span class="n">param</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=-</span><span class="n">step_size</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-32'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-32'>#</a>
|
|
</div>
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<h3>Plot $r_t$ against $t$ for various $\beta_2$</h3>
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<p><img alt="Plot of r_t" src="radam_r_t.png" /></p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">266</span><span class="k">def</span> <span class="nf">_test_rectification_term</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-33'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-33'>#</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">272</span> <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
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<span class="lineno">273</span> <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
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<span class="lineno">274</span>
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<span class="lineno">275</span> <span class="n">beta2</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.9999</span><span class="p">,</span> <span class="mf">0.999</span><span class="p">,</span> <span class="mf">0.99</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">]</span>
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<span class="lineno">276</span> <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5_000</span><span class="p">),</span> <span class="p">[[</span><span class="n">RAdam</span><span class="o">.</span><span class="n">calc_rectification_term</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">beta2</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="mi">1</span><span class="p">,</span> <span class="mi">5_000</span><span class="p">)])</span>
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<span class="lineno">277</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">beta2</span><span class="p">)</span>
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<span class="lineno">278</span> <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Optimizer"</span><span class="p">)</span>
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<span class="lineno">279</span> <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<span class="lineno">280</span>
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<span class="lineno">281</span>
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<span class="lineno">282</span><span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
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<span class="lineno">283</span> <span class="n">_test_rectification_term</span><span class="p">()</span></pre></div>
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