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<!-- navigation toc: --> <li><a href="._week40-bs001.html#___sec0" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs003.html#___sec2" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
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<!-- navigation toc: --> <li><a href="#___sec8" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs029.html#___sec28" style="font-size: 80%;"><b>Neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs043.html#___sec42" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
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<h2 id="___sec8" class="anchor">Slightly different approach </h2>
<p>
Another approach is to let the step length \( \gamma_j \) depend on the
number of epochs in such a way that it becomes very small after a
reasonable time such that we do not move at all.
<p>
As an example, let \( e = 0,1,2,3,\cdots \) denote the current epoch and let \( t_0, t_1 > 0 \) be two fixed numbers. Furthermore, let \( t = e \cdot m + i \) where \( m \) is the number of minibatches and \( i=0,\cdots,m-1 \). Then the function $$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length \( \gamma_j (0; t_0, t_1) = t_0/t_1 \) which decays in <em>time</em> \( t \).
<p>
In this way we can fix the number of epochs, compute \( \beta \) and
evaluate the cost function at the end. Repeating the computation will
give a different result since the scheme is random by design. Then we
pick the final \( \beta \) that gives the lowest value of the cost
function.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">step_length</span>(t,t0,t1):
<span style="color: #008000; font-weight: bold">return</span> t0<span style="color: #666666">/</span>(t<span style="color: #666666">+</span>t1)
n <span style="color: #666666">=</span> <span style="color: #666666">100</span> <span style="color: #408080; font-style: italic">#100 datapoints </span>
M <span style="color: #666666">=</span> <span style="color: #666666">5</span> <span style="color: #408080; font-style: italic">#size of each minibatch</span>
m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
n_epochs <span style="color: #666666">=</span> <span style="color: #666666">500</span> <span style="color: #408080; font-style: italic">#number of epochs</span>
t0 <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
t1 <span style="color: #666666">=</span> <span style="color: #666666">10</span>
gamma_j <span style="color: #666666">=</span> t0<span style="color: #666666">/</span>t1
j <span style="color: #666666">=</span> <span style="color: #666666">0</span>
<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,n_epochs<span style="color: #666666">+1</span>):
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
k <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m) <span style="color: #408080; font-style: italic">#Pick the k-th minibatch at random</span>
<span style="color: #408080; font-style: italic">#Compute the gradient using the data in minibatch Bk</span>
<span style="color: #408080; font-style: italic">#Compute new suggestion for beta</span>
t <span style="color: #666666">=</span> epoch<span style="color: #666666">*</span>m<span style="color: #666666">+</span>i
gamma_j <span style="color: #666666">=</span> step_length(t,t0,t1)
j <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;gamma_j after </span><span style="color: #BB6688; font-weight: bold">%d</span><span style="color: #BA2121"> epochs: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> (n_epochs,gamma_j))
</pre></div>
<p>
<p>
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