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<h1>
[Neural Networks and Deep Learning] week3. Shallow Neural Network
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<time datetime="2017-09-19T00:00:00+02:00"><i class="fa fa-calendar"></i> Tue, 19 Sep 2017</time>
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<span class="label label-default">Series</span>
Part 3 of «Andrew Ng Deep Learning MOOC»
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#neural-networks-overview" title="Neural Networks Overview">Neural Networks Overview</a></li><li><a class="toc-href" href="#neural-network-representation" title="Neural Network Representation">Neural Network Representation</a></li><li><a class="toc-href" href="#computing-a-neural-networks-output" output'="" s="" title="Computing a Neural Network">Computing a Neural Network's Output</a></li><li><a class="toc-href" href="#vectorizing-across-multiple-examples" title="Vectorizing across multiple examples">Vectorizing across multiple examples</a></li><li><a class="toc-href" href="#explanation-for-vectorized-implementation" title="Explanation for Vectorized Implementation">Explanation for Vectorized Implementation</a></li><li><a class="toc-href" href="#activation-functions" title="Activation functions">Activation functions</a></li><li><a class="toc-href" href="#why-do-you-need-non-linear-activation-functions" title="Why do you need non-linear activation functions?">Why do you need non-linear activation functions?</a></li><li><a class="toc-href" href="#derivatives-of-activation-functions" title="Derivatives of activation functions">Derivatives of activation functions</a><ul><li><a class="toc-href" href="#g-sigmoid" title="g = sigmoid">g = sigmoid</a></li><li><a class="toc-href" href="#g-tanh" title="g = tanh">g = tanh</a></li><li><a class="toc-href" href="#g-relu-leaky-relu" title="g = ReLU / Leaky ReLU">g = ReLU / Leaky ReLU</a></li></ul></li><li><a class="toc-href" href="#gradient-descent-for-neural-networks_1" title="Gradient descent for Neural Networks">Gradient descent for Neural Networks</a></li><li><a class="toc-href" href="#backpropagation-intuition-optional" title="Backpropagation intuition (optional)">Backpropagation intuition (optional)</a></li><li><a class="toc-href" href="#random-initialization" title="Random Initialization">Random Initialization</a></li></ul></div>
</div>
</div>
<h2 id="neural-networks-overview">Neural Networks Overview</h2>
<p>new notation: </p>
<ul>
<li>superscript <code>[i]</code> for quantities in layer i. (compared to superscript <code>(i)</code> for ith training example).</li>
<li>subscript <code>i</code> for ith unit in a layer</li>
</ul>
<h2 id="neural-network-representation">Neural Network Representation</h2>
<p>notation: </p>
<ul>
<li><code>a^[i]</code>: activation at layer i.</li>
<li>input layer: x, layer 0.</li>
<li>hidden layer</li>
<li>output layer: prediction (yhat)</li>
<li>don't count input layer as a layer</li>
</ul>
<p>a 2 layer NN:<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image.png"/></p>
<h2 id="computing-a-neural-networks-output">Computing a Neural Network's Output</h2>
<p>each node in NN: 2 step computation</p>
<ul>
<li>z = wx + b</li>
<li>a = sigmoid(z)</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image001.png"/></p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image002.png"/><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image003.png"/></p>
<p><code>z^[1]</code> = stacking <code>z[1]_i</code>s vertically
<code>a^[1]</code> = sigmoid(<code>z^[1]</code>)
vectorize computing <code>z^[1]</code>: W = <em>stacking rows of wi.T</em><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image005.png"/><br/>
W.shape = (4,3)
b.shape = (4,1)</p>
<ul>
<li>input at layer i = <code>a^[i-1]</code> (<code>x = a[0]</code>)</li>
<li>output of each layer: <code>a[i] = sigmoid(W[i] a^[i-1] + b[i])</code></li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image006.png"/></p>
<h2 id="vectorizing-across-multiple-examples">Vectorizing across multiple examples</h2>
<p>vectorize the computation acrosse m examples.
training examples: x^(1)...x^(m)<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image007.png"/><br/>
computing all yhat(i) using forloop:<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image008.png"/></p>
<p>X = <em>stacking columns of x(i)</em>, <code>X = [x(1)...x(m)]</code>
Z[1] = stacking columns of z<a href="i">1</a> = [z<a href="1">1</a>...z<a href="m">1</a>]
A = stacking columns of a(i)</p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image010.png"/><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image009.png"/><br/>
<strong>horizontal index = training example</strong> <code>^(i)</code><br/>
<strong>vertical index = nodes in layer </strong><code>_i</code><strong>/ input feature</strong><code>x_i</code>
⇒ </p>
<ul>
<li>Z[1] = W[1] * X + b[1]</li>
<li>A[1] = sigmoid(Z[1])</li>
<li>Z[2] = W[2] * A[1] + b[2]</li>
<li>A[2] = sigmoid(Z[2]) = Yhat</li>
</ul>
<h2 id="explanation-for-vectorized-implementation">Explanation for Vectorized Implementation</h2>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image012.png"/></p>
<p><strong>Recap</strong>:
stacking columns of training examples <code>x(i)</code> and activations <code>a[l](i)</code><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image013.png"/><br/>
⇒ <br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image014.png"/> </p>
<h2 id="activation-functions">Activation functions</h2>
<p>general case: <code>a = g(z)</code>, where g() is a nonlinear function.</p>
<ul>
<li>sigmoid: <code>a = 1 / (1 + exp(-z))</code></li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image018.png"/><br/>
a ∈ [0,1]</p>
<ul>
<li>tanh: <code>a = (exp(z) - exp(-z)) / (exp(z) + exp(-z))</code></li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image019.png"/><br/>
a ∈ [-1, 1] — shifted sigmoid function
⇒ data is <em>centered, learning for next layer easier</em>
<em>almost always better than sigmoid</em>, except for output layer (yhat = probability ∈[0,1])</p>
<p>downside of sigmoid and tanh: <em>slope very small when |z| is large</em> — GD slow.
⇒ ReLU</p>
<ul>
<li>ReLU <code>a = max(0, z)</code></li>
</ul>
<p>da/dz = 1 or 0
NN learns faster because slope is constant when |z| large
disadvantage: da/dz = 0 when z<0
→ leaky ReLU: small slope when z<0<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image020.png"/><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image021.png"/></p>
<p><strong>Rules of thumb</strong>:</p>
<ul>
<li>output layer: sigmoid for binary classification (output probability), <em>otherwise never use sigmoid</em></li>
<li>hidden layer: use ReLU activation by default</li>
</ul>
<h2 id="why-do-you-need-non-linear-activation-functions">Why do you need non-linear activation functions?</h2>
<p>use a linear activation function g(z) = z ?
⇒ <code>yhat</code> will just be a <em>linear function</em> of <code>x</code>. <code>yhat = Wx+b</code><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image022.png"/><br/>
one single place when using linear activation: in output layer ( y∈R )when doing regression</p>
<h2 id="derivatives-of-activation-functions">Derivatives of activation functions</h2>
<p><em>formulas for g'(z)</em></p>
<h3 id="g-sigmoid">g = sigmoid</h3>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image023.png"/><br/>
⇒ <code>g'(z) = g(z) * (1 - g(z)) = a * (1-a)</code></p>
<ul>
<li>when z = +inf, g(z) = 1, g'(z) = 1*(1-1) = 0</li>
<li>when z = -inf, g(z) = 0, g'(z) = 0</li>
<li>when z = 0, g(z) = 0.5, g'(z) = 0.25</li>
</ul>
<h3 id="g-tanh">g = tanh</h3>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image024.png"/><br/>
⇒ <code>g'(z) = 1 - tanh(z)^2 = 1 - a^2</code></p>
<ul>
<li>when z = +inf, tanh(z) = 1, g' = 0</li>
<li>when z = -inf, tanh(z) = -1, g' = 0</li>
<li>when z = 0, tanh(z) = 0, g' = 1</li>
</ul>
<h3 id="g-relu-leaky-relu">g = ReLU / Leaky ReLU</h3>
<p><strong>ReLU</strong>:
g(z) = max(0, z)
g' is <em>subgradient:</em></p>
<ul>
<li>g' = 0 when z<0</li>
<li>g' = 1 when z>=0</li>
</ul>
<p><strong>Leaky ReLU</strong>:
g(z) = max(0.01z, z)</p>
<ul>
<li>g' = 0.01 when z<0</li>
<li>g' = 1 when z>=0</li>
</ul>
<h2 id="gradient-descent-for-neural-networks_1">Gradient descent for Neural Networks</h2>
<p>NN with single hidden layer: n[0] = nx, n[1] = hidden layer size, n[2] = 1
params: w[1], b[1], w[2], b[2]</p>
<ul>
<li><code>w[1].shape=(n[1], n[0]), b[1].shape=(n[1], 1)</code></li>
<li><code>w[2].shape=(n[2], n[1]) , b[2].shape=(n[2],1)</code></li>
<li>output: yhat = a[2]</li>
</ul>
<p>cost function J(w[1],b[1],w[2],b[2]) = 1/m * sum(L(yhat, y))</p>
<p>Gradient descent:</p>
<ul>
<li>random initialization</li>
<li>repeat:<ul>
<li>compute dw[1], db[1], dw[2], db[2]</li>
<li>w[1] := w[1] - alpha*dw[1], ...</li>
</ul>
</li>
</ul>
<p>Fwd prop:<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image026.png"/><br/>
general formular for <code>l</code>th layer:<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image034.png"/> </p>
<p>Bck prop:
computing derivatives <code>dw</code>, <code>db</code>
note: use <code>keepdims = True</code> or <code>.rehape()</code> to avoid rank-1 arraies.<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image027.png"/></p>
<h2 id="backpropagation-intuition-optional">Backpropagation intuition (optional)</h2>
<p>Derive the formulas using computation graph + chain rule.<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image028.png"/></p>
<p>gradient for a single example <code>x=x(i), y=y(i)</code>:<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image029.png"/></p>
<p>vectorized implementation for i=1,..,m:
<strong>stacking columns</strong>:<code>X = [x(1),..,x(m)]</code>, <code>Z = [z(1)...z(m)]</code>, <code>Y = [y(1)..y(m)]</code>,
→ <br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image031.png"/> </p>
<h2 id="random-initialization">Random Initialization</h2>
<p>Unlike logistic regression, needs init params randomly.</p>
<p>If we init all <code>w</code> and <code>b</code> to zeros: all activations <code>a_i</code> and <code>a_j</code> will be equal → <code>dz_i = dz_j</code> → <em>all hidden units completely identical</em>
⇒ needs to init all params <em>random, small</em> number (small because we want have larger derivatives for sigmoid, which is at small values, to speed up gd).
when w is init to small rand, b don't need random init.<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk3/pasted_image033.png"/></p>
</div>
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<div id="toc"><ul><li><a class="toc-href" href="#neural-networks-overview" title="Neural Networks Overview">Neural Networks Overview</a></li><li><a class="toc-href" href="#neural-network-representation" title="Neural Network Representation">Neural Network Representation</a></li><li><a class="toc-href" href="#computing-a-neural-networks-output" output'="" s="" title="Computing a Neural Network">Computing a Neural Network's Output</a></li><li><a class="toc-href" href="#vectorizing-across-multiple-examples" title="Vectorizing across multiple examples">Vectorizing across multiple examples</a></li><li><a class="toc-href" href="#explanation-for-vectorized-implementation" title="Explanation for Vectorized Implementation">Explanation for Vectorized Implementation</a></li><li><a class="toc-href" href="#activation-functions" title="Activation functions">Activation functions</a></li><li><a class="toc-href" href="#why-do-you-need-non-linear-activation-functions" title="Why do you need non-linear activation functions?">Why do you need non-linear activation functions?</a></li><li><a class="toc-href" href="#derivatives-of-activation-functions" title="Derivatives of activation functions">Derivatives of activation functions</a><ul><li><a class="toc-href" href="#g-sigmoid" title="g = sigmoid">g = sigmoid</a></li><li><a class="toc-href" href="#g-tanh" title="g = tanh">g = tanh</a></li><li><a class="toc-href" href="#g-relu-leaky-relu" title="g = ReLU / Leaky ReLU">g = ReLU / Leaky ReLU</a></li></ul></li><li><a class="toc-href" href="#gradient-descent-for-neural-networks_1" title="Gradient descent for Neural Networks">Gradient descent for Neural Networks</a></li><li><a class="toc-href" href="#backpropagation-intuition-optional" title="Backpropagation intuition (optional)">Backpropagation intuition (optional)</a></li><li><a class="toc-href" href="#random-initialization" title="Random Initialization">Random Initialization</a></li></ul></div>
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