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<!doctype html>
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<section>
<p>
<h1>Stochastic Gradient Descent</h1>
<h2>Selected topics in Deep Learning</h2>
<h3>in 6 hours or less</h3>
<h3>by Sergey Plis</h3>
<p>
</section>
</section>
<!-- ------------------------------------------------------------------------- -->
<section>
<section data-background="figures/stochastic_grain.gif">
<h2 style="text-shadow: 4px 4px 4px #002b36; color: #93a1a1">Stochastic Gradient Descent</h2>
<div class="slide-footer">
heavily based on Sebastian Ruder's slides for <a href="https://arxiv.org/abs/1609.04747">An overview of gradient descent optimization algorithms</a>
</div>
</section>
<section>
<h2>Gradient Descent</h2>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width: 100%;">
A way to minimize an objective function $\prob{J}{\theta}$
</blockquote>
<ul>
<li> $\prob{J}{\theta}$: Objective function
<li> $\theta \in \RR^d$: parameters of the model
<li> $\eta$: Learning rate that determines the size of steps we take
</ul>
<row>
<col50>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px;width: 100%;">
Update Equation
</blockquote>
<blockquote style="background-color: #eee8d5; width: 100%;">
$\theta = \theta - \eta \nabla_{\theta} \prob{J}{\theta}$
</blockquote>
</col50>
<col50>
<img style="border:0; box-shadow: 0px 0px 0px rgba(150, 150, 255, 1); " width="1000"
src="figures/SGD_minimum.svg" alt="Gradient Descent">
</col50>
</row>
</section>
<section>
<h2>Gradient Descent Variants</h2>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width: 100%;">
There are 3 of them
</blockquote>
<ul>
<li class="fragment roll-in"> Batch gradient descent
<li class="fragment roll-in"> Stochastic gradient descent
<li class="fragment roll-in"> Mini-batch gradient descent
</ul>
<br>
<br>
<row>
<col50>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width: 100%;">
Update Equation
</blockquote>
<blockquote style="background-color: #eee8d5; width: 100%;">
$\theta = \theta - \eta {\color{red} \nabla_{\theta} \prob{J}{\theta}}$
</blockquote>
</col50>
<col50>
<blockquote style="background-color: #eee8d5; width: 90%;">
<alert>The red term</alert> is different for each method
</blockquote>
</col50>
</row>
</section>
<section>
<h2>Trade Off</h2>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px;">
Depending on the amount of data
</blockquote>
<ul>
<li> The accuracy of the parameter update
<li> The time is takes to perform an update
</ul>
<br>
<br>
<table>
<tr>
<th> Method </th>
<th> Accuracy </th>
<th> Time </th>
<th> Memory </th>
<th> Online </th>
</tr>
<tr>
<td>Batch</td>
<td><span class="far fa-thumbs-up"/></td>
<td>slow</td>
<td>high</td>
<td><span class="fa fa-times"/></td>
</tr>
<tr>
<td>Stochastic</td>
<td><i class="far fa-thumbs-down"></i></td>
<td>fast</td>
<td>low</td>
<td><span class="far fa-check-circle"/></td>
</tr>
<tr>
<td>Mini-batch</td>
<td><span class="far fa-thumbs-up"/></td>
<td>moderate</td>
<td>moderate</td>
<td><span class="far fa-check-circle"/></td>
</tr>
</table>
</section>
<section>
<h2>Batch gradient descent</h2>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width:100%;">
Compute the gradient of $\prob{J}{\theta}$ with respect to the entire dataset
</blockquote>
<row>
<col50>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px;">
Update Equation
</blockquote>
<blockquote style="background-color: #eee8d5;">
$\theta = \theta - \eta \nabla_{\theta} \prob{J}{\theta}$
</blockquote>
</col50>
<col50>
<blockquote style="background-color: #eee8d5; width: 100%;">
We need to calculate the gradients for the whole dataset to perform <b>just one update</b>.
</blockquote>
</col50>
</row>
<pre><code>
for i in range(number_of_epochs):
gradient = eval_gradient(loss_fun, data, parameters)
parameters = parameters - eta * gradient
</code></pre>
</section>
<section>
<h2>Batch gradient descent</h2>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width: 100%;" class="fragment" data-fragment-index="0">
Advantages
</blockquote>
<ul style="font-size: 36px;">
<li class="fragment roll-in" data-fragment-index="1"> We're working with the best possible error surface
<li class="fragment roll-in" data-fragment-index="2"> Guaranteed to converge to a local or global minimum of <b>that</b> surface
</ul>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width: 100%;" class="fragment" data-fragment-index="3">
Disadvantages
</blockquote>
<ul style="font-size: 36px;">
<li class="fragment roll-in" data-fragment-index="4"> Can be very slow
<li class="fragment roll-in" data-fragment-index="5"> Can be intractable due to memory requirements
<li class="fragment roll-in" data-fragment-index="6"> No online updates
</ul>
</section>
<section>
<h2>Stochastic gradient descent</h2>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width:100%;">
Perform a parameter update for each training example $\vec{x}_i$ and the corresponding label $y_i$
</blockquote>
<row>
<col50>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px;width: 100%; ">
Update Equation
</blockquote>
<blockquote style="background-color: #eee8d5; width: 100%;">
$\theta = \theta - \eta \nabla_{\theta} \prob{J}{\theta; \vec{x}_i; y_i}$
</blockquote>
</col50>
<col50>
<blockquote style="background-color: #eee8d5; width: 90%;">
We need to evaluate the gradient <b>only for a single data sample</b>.
</blockquote>
</col50>
</row>
<pre><code data-trim data-noescape>
for i in range(number_of_epochs):
np.random.shuffle(data)
for example in data:
gradient = eval_gradient(loss_fun, example, parameters)
parameters = parameters - eta * gradient
</code></pre>
</section>
<section>
<h2>Stochastic gradient descent</h2>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width: 100%;" class="fragment" data-fragment-index="0">
Advantages
</blockquote>
<ul style="font-size: 36px;">
<li class="fragment roll-in" data-fragment-index="1"> It is usually much faster than batch gradient descent.
<li class="fragment roll-in" data-fragment-index="2"> It can be used to learn online.
</ul>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width: 100%;" class="fragment" data-fragment-index="3">
Disadvantages
</blockquote>
<ul style="font-size: 36px;">
<li class="fragment roll-in" data-fragment-index="4"> It performs frequent updates with a high variance that cause the objective function to fluctuate heavily.
</ul>
</section>
<section>
<h2>The fluctuations</h2>
<row>
<col50>
<ul style="font-size: 28px;">
<li class="fragment roll-in"> Batch gradient descent converges to the minimum of the basin the parameters are placed in and the fluctuation is small.
<li class="fragment roll-in"> SGD’s fluctuation is large but it enables to jump to new and potentially better local minima.
<li class="fragment roll-in"> However, this ultimately complicates convergence to the exact minimum, as SGD will keep overshooting
</ul>
</col50>
<col50>
<img width="100%"
src="figures/SGD_fluctuations.svg" alt="Fluctuations">
</col50>
</row>
</section>
<section>
<h2>learning rate annealing</h2>
<blockquote style="background-color: #eee8d5;">
When we slowly decrease the learning rate, SGD shows the same convergence behaviour as batch gradient descent
</blockquote>
</section>
<section>
<h2>Mini-batch gradient descent</h2>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width:100%;">
Perform an update on a small sample of data
</blockquote>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px;">
Update Equation
</blockquote>
<blockquote style="background-color: #eee8d5;">
<center>
$\theta = \theta - \eta \nabla_{\theta} \prob{J}{\theta; \vec{x}_{i:i+n}; y_{i:i+n}}$
</center>
</blockquote>
<pre><code data-trim data-noescape>
for i in range(number_of_epochs):
np.random.shuffle(data)
for batch in batch_iterator(data, batch_size=32):
gradient = eval_gradient(loss_fun, batch, parameters)
parameters = parameters - eta * gradient
</code></pre>
</section>
<section>
<h2>Mini-batch gradient descent</h2>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width: 100%;" class="fragment" data-fragment-index="0">
Advantages
</blockquote>
<ul style="font-size: 36px;">
<li class="fragment roll-in" data-fragment-index="1"> Reduces the variance of the parameter updates.
<li class="fragment roll-in" data-fragment-index="2"> Can make use of highly optimized matrix optimizations common to deep learning libraries that make computing the gradient very efficiently.
</ul>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width: 100%;" class="fragment" data-fragment-index="3">
Disadvantages
</blockquote>
<ul style="font-size: 36px;">
<li class="fragment roll-in" data-fragment-index="4"> We have to set the mini-batch size
</ul>
</section>
<section>
<h2>Challenges</h2>
<ul style="font-size: 36px;">
<li class="fragment roll-in"> How to set the learning rate
<li class="fragment roll-in"> How to set the learning rate schedule
<li class="fragment roll-in"> How to change the learning rate per parameter
<li class="fragment roll-in"> How to avoid local minima and saddle points
</ul>
</section>
<section>
<h3>Gradient descent optimization algorithms</h3>
<ul>
<li> Momentum
<li> Nesterov accelerated gradient
<li> Adagrad
<li> RMSprop
<li> Adadelta
<li> Adam
</ul>
</section>
<section>
<h2>SGD problems</h2>
<img width="800" src="figures/SGD_ravines.png" alt="Ravines"><br>
<a href="https://distill.pub/2017/momentum/">Let's look at a simple demo</a>
</section>
<section data-background="figures/momentum.gif">
</section>
<section>
<h2>Momentum</h2>
<blockquote style="background-color: #eee8d5;">
\begin{align*}
\vec{v}_t & = \gamma \vec{v}_{t-1} + \eta \nabla_{\theta} \prob{J}{\theta}\\
\vec{\theta} &= \vec{\theta} - \vec{v}_t \\
\gamma &\simeq 0.9
\end{align*}
</blockquote>
<div class="fragment" data-fragment-index="0" >
Momentum is accumulated the farther the ball rolls downhill
<br>
<a href="https://distill.pub/2017/momentum/">Let's see the demo again</a>
</div>
</section>
<section>
<h2>Nesterov accelerated gradient</h2>
<h3>when we want to do better than that</h3>
<blockquote style="background-color: #eee8d5;">
\begin{align*}
\vec{v}_t & = \gamma \vec{v}_{t-1} + \eta \nabla_{\theta} \prob{J}{\theta - \gamma \vec{v}_{t-1}}\\
\vec{\theta} &= \vec{\theta} - \vec{v}_t \\
\gamma &\simeq 0.9
\end{align*}
</blockquote>
</section>
<section>
<h2>AdaGrad</h2>
<h3>we want parameter-specific learning rate </h3>
<blockquote style="background-color: #eee8d5;">
\begin{align*}
\vec{v}_t & = \vec{v}_{t-1} + (\nabla_{\theta} \prob{J}{\vec{\theta}})^2\\
\vec{\theta} &= \vec{\theta} - \frac{\eta}{\sqrt{{\prob{diag}{\vec{v}_t} + \epsilon}}} \nabla_{\theta} \prob{J}{\vec{\theta}} \\
\end{align*}
</blockquote>
</section>
<section>
<h3>RMSprop:
stop diminishing learning rate</h3>
<blockquote style="background-color: #eee8d5; width: 100%" class="fragment" data-fragment-index="0" >
\begin{align*}
\vec{g}_t & = \nabla_{\theta} \prob{J}{\vec{\theta}} \\
\prob{E}{\vec{g}^2}_t & = \gamma\prob{E}{\vec{g}^2}_{t-1} + (1-\gamma)\vec{g}_t^2\\
\vec{\theta} &= \vec{\theta} - \frac{\eta}{\sqrt{{\prob{diag}{\prob{E}{\vec{g}^2}_t} + \epsilon}}} \vec{g}_t \\
\end{align*}
</blockquote>
<blockquote style="background-color: #eee8d5; width: 100%; font-size: 32px;" class="fragment" data-fragment-index="1" >
\begin{align*}
\mbox{units of}\, \Delta\vec{\theta} \propto \mbox{units of } \vec{g} \propto \mbox{units of } \frac{\partial J}{\partial \vec{\theta}}\propto \frac{1}{\mbox{units of}\, \vec{\theta}}\\
\mbox{but} \quad \Delta\vec{\theta} \propto \mathbf{H}^{-1}\vec{g} \propto \frac{ \frac{\partial J}{\partial \vec{\theta}}}{ \frac{\partial^2 J}{\partial \vec{\theta}^2}} \propto \mbox{units of }\, \vec{\theta}
\end{align*}
</blockquote>
</section>
<section>
<h3>Adadelta:
get everything right</h3>
<blockquote style="background-color: #eee8d5; width: 100%" >
\begin{align*}
\vec{g}_t & = \nabla_{\theta} \prob{J}{\vec{\theta}} \\
\Delta\vec{\theta}_{t-1} & = \frac{\eta}{\sqrt{{\prob{diag}{\prob{E}{\vec{g}^2}_{t-1}} + \epsilon}}} \\
\prob{E}{\Delta\vec{\theta}^2}_{t-1} & = \gamma\prob{E}{\Delta\vec{\theta}^2}_{t-2} + (1-\gamma)\Delta\vec{\theta}^2_{t-1}\\
\prob{E}{\vec{g}^2}_t & = \gamma\prob{E}{\vec{g}^2}_{t-1} + (1-\gamma)\vec{g}_t^2\\
\vec{\theta} &= \vec{\theta} - \frac{{\sqrt{{\prob{diag}{\prob{E}{\Delta\vec{\theta}^2}_{t-1}} + \epsilon}}} }{{\sqrt{{\prob{diag}{\prob{E}{\vec{g}^2}_{t}} + \epsilon}}} }\vec{g}_t \\
\end{align*}
</blockquote>
</section>
<section>
<h2>Adam (adaptive moment)</h2>
<blockquote style="background-color: #eee8d5; font-size:32px">
\begin{align*}
\vec{m}_t & = \beta_1 \vec{m}_{t-1} + (1 - \beta_1) \nabla_{\theta} \prob{J}{\theta}\\
\vec{v}_t & = \beta_2 \vec{v}_{t-1} + (1 - \beta_2) (\nabla_{\theta} \prob{J}{\theta})^2\\
\hat{\vec{m}}_t &= \frac{\vec{m}_t}{1 - \beta_1^t}\\
\hat{\vec{v}}_t &= \frac{\vec{v}_t}{1 - \beta_2^t}\\
\vec{\theta}_{t+1} &= \vec{\theta}_t - \frac{\eta}{\sqrt{\hat{\vec{v}}_t} + \epsilon} \hat{\vec{m}}_t\\
\beta_1 &\simeq 0.9 \quad
\beta_2 \simeq 0.999 \quad
\epsilon \simeq 10^{-8}
\end{align*}
</blockquote>
</section>
<section>
<h2>Many more</h2>
<ul>
<li> AdaMax
<li> Nadam
<li> AMSgrad
<li> AdamW
<li> QHAdam
<li> AggMo
</ul>
<h2>but use Adam if in doubt</h2>
</section>
<section data-background="figures/sgd_approaches.gif" data-background-size="contain">
</section>
<section>
<h2>The rise of the SGD</h2>
<img width="1000" src="figures/sgd_table.png" alt="Ravines"><br>
<div class="slide-footer"><a href="https://ai.googleblog.com/2018/12/the-neurips-2018-test-of-time-award.html">
"The Trade-Offs of Large Scale Learning" by Léon Bottou and Olivier Bousquet 2007
</a><br>
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Talk about why and how SGD became
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