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Varuna Jayasiri
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docs/normalization/batch_norm/index.html
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@ -85,8 +85,8 @@ could be in distribution $\mathcal{N}(0.5, 1)$.
Then, after some training steps, it could move to $\mathcal{N}(0.5, 1)$.
This is <em>internal covariate shift</em>.</p>
<p>Internal covariate shift will adversely affect training speed because the later layers
($l_2$ in the above example) has to adapt to this shifted distribution.</p>
<p>By stabilizing the distribution batch normalization minimizes the internal covariate shift.</p>
($l_2$ in the above example) have to adapt to this shifted distribution.</p>
<p>By stabilizing the distribution, batch normalization minimizes the internal covariate shift.</p>
<h2>Normalization</h2>
<p>It is known that whitening improves training speed and convergence.
<em>Whitening</em> is linearly transforming inputs to have zero mean, unit variance,
@ -95,9 +95,9 @@ and be uncorrelated.</p>
<p>Normalizing outside the gradient computation using pre-computed (detached)
means and variances doesn&rsquo;t work. For instance. (ignoring variance), let
<script type="math/tex; mode=display">\hat{x} = x - \mathbb{E}[x]</script>
where $x = u + b$ and $b$ is a trained bias.
and $\mathbb{E}[x]$ is outside gradient computation (pre-computed constant).</p>
<p>Note that $\hat{x}$ has no effect of $b$.
where $x = u + b$ and $b$ is a trained bias
and $\mathbb{E}[x]$ is an outside gradient computation (pre-computed constant).</p>
<p>Note that $\hat{x}$ has no effect on $b$.
Therefore,
$b$ will increase or decrease based
$\frac{\partial{\mathcal{L}}}{\partial x}$,
@ -106,14 +106,14 @@ The paper notes that similar explosions happen with variances.</p>
<h3>Batch Normalization</h3>
<p>Whitening is computationally expensive because you need to de-correlate and
the gradients must flow through the full whitening calculation.</p>
<p>The paper introduces simplified version which they call <em>Batch Normalization</em>.
<p>The paper introduces a simplified version which they call <em>Batch Normalization</em>.
First simplification is that it normalizes each feature independently to have
zero mean and unit variance:
<script type="math/tex; mode=display">\hat{x}^{(k)} = \frac{x^{(k)} - \mathbb{E}[x^{(k)}]}{\sqrt{Var[x^{(k)}]}}</script>
where $x = (x^{(1)} &hellip; x^{(d)})$ is the $d$-dimensional input.</p>
<p>The second simplification is to use estimates of mean $\mathbb{E}[x^{(k)}]$
and variance $Var[x^{(k)}]$ from the mini-batch
for normalization; instead of calculating the mean and variance across whole dataset.</p>
for normalization; instead of calculating the mean and variance across the whole dataset.</p>
<p>Normalizing each feature to zero mean and unit variance could affect what the layer
can represent.
As an example paper illustrates that, if the inputs to a sigmoid are normalized
@ -126,8 +126,8 @@ where $y^{(k)}$ is the output of the batch normalization layer.</p>
like $Wu + b$ the bias parameter $b$ gets cancelled due to normalization.
So you can and should omit bias parameter in linear transforms right before the
batch normalization.</p>
<p>Batch normalization also makes the back propagation invariant to the scale of the weights.
And empirically it improves generalization, so it has regularization effects too.</p>
<p>Batch normalization also makes the back propagation invariant to the scale of the weights
and empirically it improves generalization, so it has regularization effects too.</p>
<h2>Inference</h2>
<p>We need to know $\mathbb{E}[x^{(k)}]$ and $Var[x^{(k)}]$ in order to
perform the normalization.
@ -136,7 +136,7 @@ and find the mean and variance, or you can use an estimate calculated during tra
The usual practice is to calculate an exponential moving average of
mean and variance during the training phase and use that for inference.</p>
<p>Here&rsquo;s <a href="mnist.html">the training code</a> and a notebook for training
a CNN classifier that use batch normalization for MNIST dataset.</p>
a CNN classifier that uses batch normalization for MNIST dataset.</p>
<p><a href="https://colab.research.google.com/github/lab-ml/nn/blob/master/labml_nn/normalization/batch_norm/mnist.ipynb"><img alt="Open In Colab" src="https://colab.research.google.com/assets/colab-badge.svg" /></a>
<a href="https://web.lab-ml.com/run?uuid=011254fe647011ebbb8e0242ac1c0002"><img alt="View Run" src="https://img.shields.io/badge/labml-experiment-brightgreen" /></a></p>
</div>
@ -251,7 +251,7 @@ mean $\mathbb{E}[x^{(k)}]$ and variance $Var[x^{(k)}]$</p>
<a href='#section-6'>#</a>
</div>
<p><code>x</code> is a tensor of shape <code>[batch_size, channels, *]</code>.
<code>*</code> could be any number of (even 0) dimensions.
<code>*</code> denotes any number of (possibly 0) dimensions.
For example, in an image (2D) convolution this will be
<code>[batch_size, channels, height, width]</code></p>
</div>
@ -286,7 +286,7 @@ mean $\mathbb{E}[x^{(k)}]$ and variance $Var[x^{(k)}]$</p>
<div class='section-link'>
<a href='#section-9'>#</a>
</div>
<p>Sanity check to make sure the number of features is same</p>
<p>Sanity check to make sure the number of features is the same</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">174</span> <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">channels</span> <span class="o">==</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span></pre></div>