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  <div class="section" id="optimizer">
<h1>Optimizer<a class="headerlink" href="#optimizer" title="Permalink to this headline">¶</a></h1>
<div class="section" id="momentum">
<h2>Momentum<a class="headerlink" href="#momentum" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt>
<em class="property">class </em><code class="descclassname">paddle.v2.optimizer.</code><code class="descname">Momentum</code><span class="sig-paren">(</span><em>momentum=None</em>, <em>sparse=False</em>, <em>**kwargs</em><span class="sig-paren">)</span></dt>
<dd><p>Momentum Optimizer.</p>
<p>When sparse=False, the momentum update formula is as follows:</p>
<div class="math">
\[\begin{split}v_{t} &amp;= k * v_{t-1} - \gamma_t (g_{t} + \lambda w_{t-1}) \\
w_{t} &amp;= w_{t-1} + v_{t} \\\end{split}\]</div>
<p>where, <span class="math">\(k\)</span> is momentum, <span class="math">\(\lambda\)</span> is decay rate,
<span class="math">\(\gamma_t\)</span> is learning rate at the t&#8217;th iteration.
<span class="math">\(w_{t}\)</span> is the weight as the t&#8217;th iteration.
And the <span class="math">\(v_{t}\)</span> is the history momentum variable.</p>
<p>When sparse=True, the update scheme:</p>
<div class="math">
\[\begin{split}\alpha_t &amp;= \alpha_{t-1} / k \\
\beta_t &amp;= \beta_{t-1} / (1 + \lambda \gamma_t) \\
u_t &amp;= u_{t-1} - \alpha_t \gamma_t g_t \\
v_t &amp;= v_{t-1} + \tau_{t-1} \alpha_t \gamma_t g_t \\
\tau_t &amp;= \tau_{t-1} + \beta_t / \alpha_t\end{split}\]</div>
<p>where <span class="math">\(k\)</span> is momentum, <span class="math">\(\lambda\)</span> is decay rate,
<span class="math">\(\gamma_t\)</span> is learning rate at the t&#8217;th iteration.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
<li><strong>momentum</strong> (<em>float</em>) &#8211; the momentum factor.</li>
<li><strong>sparse</strong> (<em>bool</em>) &#8211; with sparse support or not, False by default.</li>
</ul>
</td>
</tr>
</tbody>
</table>
</dd></dl>

</div>
<div class="section" id="adam">
<h2>Adam<a class="headerlink" href="#adam" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt>
<em class="property">class </em><code class="descclassname">paddle.v2.optimizer.</code><code class="descname">Adam</code><span class="sig-paren">(</span><em>beta1=0.9</em>, <em>beta2=0.999</em>, <em>epsilon=1e-08</em>, <em>**kwargs</em><span class="sig-paren">)</span></dt>
<dd><p>Adam optimizer.
The details of please refer <a class="reference external" href="https://arxiv.org/abs/1412.6980">Adam: A Method for Stochastic Optimization</a></p>
<div class="math">
\[\begin{split}m(w, t) &amp; = \beta_1 m(w, t-1) + (1 - \beta_1) \nabla Q_i(w) \\
v(w, t) &amp; = \beta_2 v(w, t-1) + (1 - \beta_2)(\nabla Q_i(w)) ^2 \\
w &amp; = w - \frac{\eta m(w, t)}{\sqrt{v(w,t) + \epsilon}}\end{split}\]</div>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
<li><strong>beta1</strong> (<em>float</em>) &#8211; the <span class="math">\(\beta_1\)</span> in equation.</li>
<li><strong>beta2</strong> (<em>float</em>) &#8211; the <span class="math">\(\beta_2\)</span> in equation.</li>
<li><strong>epsilon</strong> (<em>float</em>) &#8211; the <span class="math">\(\epsilon\)</span> in equation. It is used to prevent
divided by zero.</li>
</ul>
</td>
</tr>
</tbody>
</table>
</dd></dl>

</div>
<div class="section" id="adamax">
<h2>Adamax<a class="headerlink" href="#adamax" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt>
<em class="property">class </em><code class="descclassname">paddle.v2.optimizer.</code><code class="descname">Adamax</code><span class="sig-paren">(</span><em>beta1=0.9</em>, <em>beta2=0.999</em>, <em>**kwargs</em><span class="sig-paren">)</span></dt>
<dd><p>Adamax optimizer.</p>
<p>The details of please refer this <a class="reference external" href="https://arxiv.org/abs/1412.6980">Adam: A Method for Stochastic Optimization</a></p>
<div class="math">
\[\begin{split}m_t &amp; = \beta_1 * m_{t-1} + (1-\beta_1)* \nabla Q_i(w) \\
u_t &amp; = max(\beta_2*u_{t-1}, abs(\nabla Q_i(w))) \\
w_t &amp; = w_{t-1} - (\eta/(1-\beta_1^t))*m_t/u_t\end{split}\]</div>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
<li><strong>beta1</strong> (<em>float</em>) &#8211; the <span class="math">\(\beta_1\)</span> in the equation.</li>
<li><strong>beta2</strong> (<em>float</em>) &#8211; the <span class="math">\(\beta_2\)</span> in the equation.</li>
</ul>
</td>
</tr>
</tbody>
</table>
</dd></dl>

</div>
<div class="section" id="adagrad">
<h2>AdaGrad<a class="headerlink" href="#adagrad" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt>
<em class="property">class </em><code class="descclassname">paddle.v2.optimizer.</code><code class="descname">AdaGrad</code><span class="sig-paren">(</span><em>**kwargs</em><span class="sig-paren">)</span></dt>
<dd><p>Adagrad(for ADAptive GRAdient algorithm) optimizer.</p>
<p>For details please refer this <a class="reference external" href="http://www.magicbroom.info/Papers/DuchiHaSi10.pdf">Adaptive Subgradient Methods for
Online Learning and Stochastic Optimization</a>.</p>
<div class="math">
\[\begin{split}G &amp;= \sum_{\tau=1}^{t} g_{\tau} g_{\tau}^T \\
w &amp; = w - \eta diag(G)^{-\frac{1}{2}} \circ g\end{split}\]</div>
</dd></dl>

</div>
<div class="section" id="decayedadagrad">
<h2>DecayedAdaGrad<a class="headerlink" href="#decayedadagrad" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt>
<em class="property">class </em><code class="descclassname">paddle.v2.optimizer.</code><code class="descname">DecayedAdaGrad</code><span class="sig-paren">(</span><em>rho=0.95</em>, <em>epsilon=1e-06</em>, <em>**kwargs</em><span class="sig-paren">)</span></dt>
<dd><p>AdaGrad method with decayed sum gradients. The equations of this method
show as follow.</p>
<div class="math">
\[\begin{split}E(g_t^2) &amp;= \rho * E(g_{t-1}^2) + (1-\rho) * g^2 \\
learning\_rate &amp;= 1/sqrt( ( E(g_t^2) + \epsilon )\end{split}\]</div>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
<li><strong>rho</strong> (<em>float</em>) &#8211; The <span class="math">\(\rho\)</span> parameter in that equation</li>
<li><strong>epsilon</strong> (<em>float</em>) &#8211; The <span class="math">\(\epsilon\)</span> parameter in that equation.</li>
</ul>
</td>
</tr>
</tbody>
</table>
</dd></dl>

</div>
<div class="section" id="adadelta">
<h2>AdaDelta<a class="headerlink" href="#adadelta" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt>
<em class="property">class </em><code class="descclassname">paddle.v2.optimizer.</code><code class="descname">AdaDelta</code><span class="sig-paren">(</span><em>rho=0.95</em>, <em>epsilon=1e-06</em>, <em>**kwargs</em><span class="sig-paren">)</span></dt>
<dd><p>AdaDelta method. The details of adadelta please refer to this
<a class="reference external" href="http://www.matthewzeiler.com/pubs/googleTR2012/googleTR2012.pdf">ADADELTA: AN ADAPTIVE LEARNING RATE METHOD</a>.</p>
<div class="math">
\[\begin{split}E(g_t^2) &amp;= \rho * E(g_{t-1}^2) + (1-\rho) * g^2 \\
learning\_rate &amp;= sqrt( ( E(dx_{t-1}^2) + \epsilon ) / ( \
                  E(g_t^2) + \epsilon ) ) \\
E(dx_t^2) &amp;= \rho * E(dx_{t-1}^2) + (1-\rho) * (-g*learning\_rate)^2\end{split}\]</div>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
<li><strong>rho</strong> (<em>float</em>) &#8211; <span class="math">\(\rho\)</span> in equation</li>
<li><strong>epsilon</strong> (<em>float</em>) &#8211; <span class="math">\(\rho\)</span> in equation</li>
</ul>
</td>
</tr>
</tbody>
</table>
</dd></dl>

</div>
<div class="section" id="rmsprop">
<h2>RMSProp<a class="headerlink" href="#rmsprop" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt>
<em class="property">class </em><code class="descclassname">paddle.v2.optimizer.</code><code class="descname">RMSProp</code><span class="sig-paren">(</span><em>rho=0.95</em>, <em>epsilon=1e-06</em>, <em>**kwargs</em><span class="sig-paren">)</span></dt>
<dd><p>RMSProp(for Root Mean Square Propagation) optimizer. For details please
refer this <a class="reference external" href="http://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf">slide</a>.</p>
<p>The equations of this method as follows:</p>
<div class="math">
\[\begin{split}v(w, t) &amp; = \rho v(w, t-1) + (1 - \rho)(\nabla Q_{i}(w))^2 \\
w &amp; = w - \frac{\eta} {\sqrt{v(w,t) + \epsilon}} \nabla Q_{i}(w)\end{split}\]</div>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
<li><strong>rho</strong> (<em>float</em>) &#8211; the <span class="math">\(\rho\)</span> in the equation. The forgetting factor.</li>
<li><strong>epsilon</strong> (<em>float</em>) &#8211; the <span class="math">\(\epsilon\)</span> in the equation.</li>
</ul>
</td>
</tr>
</tbody>
</table>
</dd></dl>

</div>
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