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<h1>AMSGrad</h1>
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<p>This is a <a href="https://pytorch.org">PyTorch</a> implementation of the paper
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<a href="https://arxiv.org/abs/1904.09237">On the Convergence of Adam and Beyond</a>.</p>
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<p>We implement this as an extension to our <a href="adam.html">Adam optimizer implementation</a>.
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The implementation it self is really small since it’s very similar to Adam.</p>
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<p>We also have an implementation of the synthetic example described in the paper where Adam fails to converge.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">18</span><span></span><span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Dict</span>
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<span class="lineno">19</span>
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<span class="lineno">20</span><span class="kn">import</span> <span class="nn">torch</span>
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<span class="lineno">21</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">22</span>
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<span class="lineno">23</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">24</span><span class="kn">from</span> <span class="nn">labml_nn.optimizers.adam</span> <span class="kn">import</span> <span class="n">Adam</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>AMSGrad Optimizer</h2>
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<p>This class extends from Adam optimizer defined in <a href="adam.html"><code>adam.py</code></a>.
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Adam optimizer is extending the class <code>GenericAdaptiveOptimizer</code>
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defined in <a href="index.html"><code>__init__.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">27</span><span class="k">class</span> <span class="nc">AMSGrad</span><span class="p">(</span><span class="n">Adam</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>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">35</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">36</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">37</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">38</span> <span class="n">amsgrad</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">53</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">54</span> <span class="n">defaults</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">amsgrad</span><span class="o">=</span><span class="n">amsgrad</span><span class="p">))</span>
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<span class="lineno">55</span>
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<span class="lineno">56</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">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>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">58</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>Call <code>init_state</code> of Adam optimizer which we are extending</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">68</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="n">init_state</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></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>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 squared gradient</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">72</span> <span class="k">if</span> <span class="n">group</span><span class="p">[</span><span class="s1">'amsgrad'</span><span class="p">]:</span>
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<span class="lineno">73</span> <span class="n">state</span><span class="p">[</span><span class="s1">'max_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-7'>
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<div class='docs doc-strings'>
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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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<h3>Calculate $m_t$ and and $v_t$ or $\max(v_1, v_2, …, v_{t-1}, v_t)$</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">75</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="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></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>Get $m_t$ and $v_t$ from <em>Adam</em></p>
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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">m</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="nb">super</span><span class="p">()</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-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>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">88</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-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 $\max(v_1, v_2, …, v_{t-1})$.</p>
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<p>🗒 The paper uses the notation $\hat{v}_t$ for this, which we don’t use
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that here because it confuses with the Adam’s usage of the same notation
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for bias corrected exponential moving average.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">94</span> <span class="n">v_max</span> <span class="o">=</span> <span class="n">state</span><span class="p">[</span><span class="s1">'max_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>Calculate $\max(v_1, v_2, …, v_{t-1}, v_t)$.</p>
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<p>🤔 I feel you should be taking / maintaining the max of the bias corrected
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second exponential average of squared gradient.
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But this is how it’s
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<a href="https://github.com/pytorch/pytorch/blob/19f4c5110e8bcad5e7e75375194262fca0a6293a/torch/optim/functional.py#L90">implemented in PyTorch also</a>.
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I guess it doesn’t really matter since bias correction only increases the value
|
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and it only makes an actual difference during the early few steps of the training.</p>
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</div>
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<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">103</span> <span class="n">torch</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">v_max</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">out</span><span class="o">=</span><span class="n">v_max</span><span class="p">)</span>
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<span class="lineno">104</span>
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<span class="lineno">105</span> <span class="k">return</span> <span class="n">m</span><span class="p">,</span> <span class="n">v_max</span>
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<span class="lineno">106</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-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>Fall back to <em>Adam</em> if the parameter group is not 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">108</span> <span class="k">return</span> <span class="n">m</span><span class="p">,</span> <span class="n">v</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 doc-strings'>
|
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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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<h2>Synthetic Experiment</h2>
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<p>This is the synthetic experiment described in the paper,
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that shows a scenario where <em>Adam</em> fails.</p>
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<p>The paper (and Adam) formulates the problem of optimizing as
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minimizing the expected value of a function, $\mathbb{E}[f(\theta)]$
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with respect to the parameters $\theta$.
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In the stochastic training setting we do not get hold of the function $f$
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it self; that is,
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when you are optimizing a NN $f$ would be the function on entire
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batch of data.
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What we actually evaluate is a mini-batch so the actual function is
|
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realization of the stochastic $f$.
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This is why we are talking about an expected value.
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So let the function realizations be $f_1, f_2, …, f_T$ for each time step
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of training.</p>
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<p>We measure the performance of the optimizer as the regret,
|
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<script type="math/tex; mode=display">R(T) = \sum_{t=1}^T \big[ f_t(\theta_t) - f_t(\theta^*) \big]</script>
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where $theta_t$ is the parameters at time step $t$, and $\theta^*$ is the
|
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optimal parameters that minimize $\mathbb{E}[f(\theta)]$.</p>
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<p>Now lets define the synthetic problem,
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<script type="math/tex; mode=display">\begin{align}
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f_t(x) =
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\begin{cases}
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1010 x, & \text{for $t \mod 101 = 1$} \\
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-10 x, & \text{otherwise}
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\end{cases}
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\end{align}</script>
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where $-1 \le x \le +1$.
|
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The optimal solution is $x = -1$.</p>
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<p>This code will try running <em>Adam</em> and <em>AMSGrad</em> on this problem.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">111</span><span class="k">def</span> <span class="nf">_synthetic_experiment</span><span class="p">(</span><span class="n">is_adam</span><span class="p">:</span> <span class="nb">bool</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>Define $x$ parameter</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">151</span> <span class="n">x</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="o">.</span><span class="mi">0</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-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>Optimal, $x^* = -1$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">153</span> <span class="n">x_star</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span></pre></div>
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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 doc-strings'>
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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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<h3>$f_t(x)$</h3>
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</div>
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<div class='code'>
|
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<div class="highlight"><pre><span class="lineno">155</span> <span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">t</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">x_</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-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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</div>
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<div class='code'>
|
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<div class="highlight"><pre><span class="lineno">159</span> <span class="k">if</span> <span class="n">t</span> <span class="o">%</span> <span class="mi">101</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
|
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<span class="lineno">160</span> <span class="k">return</span> <span class="p">(</span><span class="mi">1010</span> <span class="o">*</span> <span class="n">x_</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
|
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<span class="lineno">161</span> <span class="k">else</span><span class="p">:</span>
|
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<span class="lineno">162</span> <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">10</span> <span class="o">*</span> <span class="n">x_</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span></pre></div>
|
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</div>
|
|
</div>
|
|
<div class='section' id='section-18'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-18'>#</a>
|
|
</div>
|
|
<p>Initialize the relevant optimizer</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">165</span> <span class="k">if</span> <span class="n">is_adam</span><span class="p">:</span>
|
|
<span class="lineno">166</span> <span class="n">optimizer</span> <span class="o">=</span> <span class="n">Adam</span><span class="p">([</span><span class="n">x</span><span class="p">],</span> <span class="n">lr</span><span class="o">=</span><span class="mf">1e-2</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">0.99</span><span class="p">))</span>
|
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<span class="lineno">167</span> <span class="k">else</span><span class="p">:</span>
|
|
<span class="lineno">168</span> <span class="n">optimizer</span> <span class="o">=</span> <span class="n">AMSGrad</span><span class="p">([</span><span class="n">x</span><span class="p">],</span> <span class="n">lr</span><span class="o">=</span><span class="mf">1e-2</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">0.99</span><span class="p">))</span></pre></div>
|
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</div>
|
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</div>
|
|
<div class='section' id='section-19'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-19'>#</a>
|
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</div>
|
|
<p>$R(T)$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">170</span> <span class="n">total_regret</span> <span class="o">=</span> <span class="mi">0</span>
|
|
<span class="lineno">171</span>
|
|
<span class="lineno">172</span> <span class="kn">from</span> <span class="nn">labml</span> <span class="kn">import</span> <span class="n">monit</span><span class="p">,</span> <span class="n">tracker</span><span class="p">,</span> <span class="n">experiment</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>Create experiment to record results</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">175</span> <span class="k">with</span> <span class="n">experiment</span><span class="o">.</span><span class="n">record</span><span class="p">(</span><span class="n">name</span><span class="o">=</span><span class="s1">'synthetic'</span><span class="p">,</span> <span class="n">comment</span><span class="o">=</span><span class="s1">'Adam'</span> <span class="k">if</span> <span class="n">is_adam</span> <span class="k">else</span> <span class="s1">'AMSGrad'</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>Run for $10^7$ steps</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">177</span> <span class="k">for</span> <span class="n">step</span> <span class="ow">in</span> <span class="n">monit</span><span class="o">.</span><span class="n">loop</span><span class="p">(</span><span class="mi">10_000_000</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>$f_t(\theta_t) - f_t(\theta^*)$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">179</span> <span class="n">regret</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">step</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">func</span><span class="p">(</span><span class="n">step</span><span class="p">,</span> <span class="n">x_star</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>$R(T) = \sum_{t=1}^T \big[ f_t(\theta_t) - f_t(\theta^*) \big]$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">181</span> <span class="n">total_regret</span> <span class="o">+=</span> <span class="n">regret</span><span class="o">.</span><span class="n">item</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>Track results every 1,000 steps</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">183</span> <span class="k">if</span> <span class="p">(</span><span class="n">step</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">1000</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
|
<span class="lineno">184</span> <span class="n">tracker</span><span class="o">.</span><span class="n">save</span><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="n">regret</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="n">regret</span><span class="o">=</span><span class="n">total_regret</span> <span class="o">/</span> <span class="p">(</span><span class="n">step</span> <span class="o">+</span> <span class="mi">1</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>Calculate gradients</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">186</span> <span class="n">regret</span><span class="o">.</span><span class="n">backward</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>Optimize</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">188</span> <span class="n">optimizer</span><span class="o">.</span><span class="n">step</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>Clear gradients</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">190</span> <span class="n">optimizer</span><span class="o">.</span><span class="n">zero_grad</span><span class="p">()</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>Make sure $-1 \le x \le +1$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">193</span> <span class="n">x</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">clamp_</span><span class="p">(</span><span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">+</span><span class="mf">1.</span><span class="p">)</span>
|
|
<span class="lineno">194</span>
|
|
<span class="lineno">195</span>
|
|
<span class="lineno">196</span><span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</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>Run the synthetic experiment is <em>Adam</em>.
|
|
<a href="https://app.labml.ai/run/61ebfdaa384411eb94d8acde48001122">Here are the results</a>.
|
|
You can see that Adam converges at $x = +1$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">200</span> <span class="n">_synthetic_experiment</span><span class="p">(</span><span class="kc">True</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>Run the synthetic experiment is <em>AMSGrad</em>
|
|
<a href="https://app.labml.ai/run/uuid=bc06405c384411eb8b82acde48001122">Here are the results</a>.
|
|
You can see that AMSGrad converges to true optimal $x = -1$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">204</span> <span class="n">_synthetic_experiment</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span></pre></div>
|
|
</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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