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<!-- navigation toc: --> <li><a href="._week41-bs002.html#___sec1" style="font-size: 80%;">Setting up the Back propagation algorithm</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs003.html#___sec2" style="font-size: 80%;">Setting up a Multi-layer perceptron model for classification</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs025.html#___sec24" style="font-size: 80%;">Visualization</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs026.html#___sec25" style="font-size: 80%;">Building neural networks in Tensorflow and Keras</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs027.html#___sec26" style="font-size: 80%;">Tensorflow</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs028.html#___sec27" style="font-size: 80%;">Using Keras</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs029.html#___sec28" style="font-size: 80%;">Collect and pre-process data</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs030.html#___sec29" style="font-size: 80%;">The Breast Cancer Data, now with Keras</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs031.html#___sec30" style="font-size: 80%;">Fine-tuning neural network hyperparameters</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs032.html#___sec31" style="font-size: 80%;">Hidden layers</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs033.html#___sec32" style="font-size: 80%;">Which activation function should I use?</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs034.html#___sec33" style="font-size: 80%;">Is the Logistic activation function (Sigmoid) our choice?</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs037.html#___sec36" style="font-size: 80%;">Which activation function should we use?</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs042.html#___sec41" style="font-size: 80%;">A very nice website on Neural Networks</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs043.html#___sec42" style="font-size: 80%;">A top-down perspective on Neural networks</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs044.html#___sec43" style="font-size: 80%;">Limitations of supervised learning with deep networks</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs045.html#___sec44" style="font-size: 80%;">Convolutional Neural Networks (recognizing images)</a></li>
<!-- navigation toc: --> <li><a href="._week41-bs046.html#___sec45" style="font-size: 80%;">Regular NNs don’t scale well to full images</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs048.html#___sec47" style="font-size: 80%;">Layers used to build CNNs</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs050.html#___sec49" style="font-size: 80%;">CNNs in brief</a></li>
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<h2 id="___sec17" class="anchor">Matrix multiplication </h2>
<p>
To more efficently train our network these equations are implemented using matrix operations.
The error in the output layer is calculated simply as, with \( \hat{t} \) being our targets,
$$ \delta_L = \hat{t} - \hat{y} = (n_{inputs}, n_{categories}) .$$
<p>
The gradient for the output weights is calculated as
$$ \nabla W_{L} = \hat{a}^T \delta_L = (n_{hidden}, n_{categories}) ,$$
<p>
where \( \hat{a} = (n_{inputs}, n_{hidden}) \). This simply means that we are summing up the gradients for each input.
Since we are going backwards we have to transpose the activation matrix.
<p>
The gradient with respect to the output bias is then
$$ \nabla \hat{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .$$
<p>
The error in the hidden layer is
$$ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,$$
<p>
where \( f'(a_{h}) \) is the derivative of the activation in the hidden layer. The matrix products mean
that we are summing up the products for each neuron in the output layer. The symbol \( \circ \) denotes
the <em>Hadamard product</em>, meaning element-wise multiplication.
<p>
This again gives us the gradients in the hidden layer:
$$ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,$$
$$ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .$$
<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: #408080; font-style: italic"># to categorical turns our integer vector into a onehot representation</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> accuracy_score
<span style="color: #408080; font-style: italic"># one-hot in numpy</span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">to_categorical_numpy</span>(integer_vector):
n_inputs <span style="color: #666666">=</span> <span style="color: #008000">len</span>(integer_vector)
n_categories <span style="color: #666666">=</span> np<span style="color: #666666">.</span>max(integer_vector) <span style="color: #666666">+</span> <span style="color: #666666">1</span>
onehot_vector <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n_inputs, n_categories))
onehot_vector[<span style="color: #008000">range</span>(n_inputs), integer_vector] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
<span style="color: #008000; font-weight: bold">return</span> onehot_vector
<span style="color: #408080; font-style: italic">#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)</span>
Y_train_onehot, Y_test_onehot <span style="color: #666666">=</span> to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">feed_forward_train</span>(X):
<span style="color: #408080; font-style: italic"># weighted sum of inputs to the hidden layer</span>
z_h <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(X, hidden_weights) <span style="color: #666666">+</span> hidden_bias
<span style="color: #408080; font-style: italic"># activation in the hidden layer</span>
a_h <span style="color: #666666">=</span> sigmoid(z_h)
<span style="color: #408080; font-style: italic"># weighted sum of inputs to the output layer</span>
z_o <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(a_h, output_weights) <span style="color: #666666">+</span> output_bias
<span style="color: #408080; font-style: italic"># softmax output</span>
<span style="color: #408080; font-style: italic"># axis 0 holds each input and axis 1 the probabilities of each category</span>
exp_term <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(z_o)
probabilities <span style="color: #666666">=</span> exp_term <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum(exp_term, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
<span style="color: #408080; font-style: italic"># for backpropagation need activations in hidden and output layers</span>
<span style="color: #008000; font-weight: bold">return</span> a_h, probabilities
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">backpropagation</span>(X, Y):
a_h, probabilities <span style="color: #666666">=</span> feed_forward_train(X)
<span style="color: #408080; font-style: italic"># error in the output layer</span>
error_output <span style="color: #666666">=</span> probabilities <span style="color: #666666">-</span> Y
<span style="color: #408080; font-style: italic"># error in the hidden layer</span>
error_hidden <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(error_output, output_weights<span style="color: #666666">.</span>T) <span style="color: #666666">*</span> a_h <span style="color: #666666">*</span> (<span style="color: #666666">1</span> <span style="color: #666666">-</span> a_h)
<span style="color: #408080; font-style: italic"># gradients for the output layer</span>
output_weights_gradient <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(a_h<span style="color: #666666">.</span>T, error_output)
output_bias_gradient <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(error_output, axis<span style="color: #666666">=0</span>)
<span style="color: #408080; font-style: italic"># gradient for the hidden layer</span>
hidden_weights_gradient <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(X<span style="color: #666666">.</span>T, error_hidden)
hidden_bias_gradient <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(error_hidden, axis<span style="color: #666666">=0</span>)
<span style="color: #008000; font-weight: bold">return</span> output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Old accuracy on training data: &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(accuracy_score(predict(X_train), Y_train)))
eta <span style="color: #666666">=</span> <span style="color: #666666">0.01</span>
lmbd <span style="color: #666666">=</span> <span style="color: #666666">0.01</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>(<span style="color: #666666">1000</span>):
<span style="color: #408080; font-style: italic"># calculate gradients</span>
dWo, dBo, dWh, dBh <span style="color: #666666">=</span> backpropagation(X_train, Y_train_onehot)
<span style="color: #408080; font-style: italic"># regularization term gradients</span>
dWo <span style="color: #666666">+=</span> lmbd <span style="color: #666666">*</span> output_weights
dWh <span style="color: #666666">+=</span> lmbd <span style="color: #666666">*</span> hidden_weights
<span style="color: #408080; font-style: italic"># update weights and biases</span>
output_weights <span style="color: #666666">-=</span> eta <span style="color: #666666">*</span> dWo
output_bias <span style="color: #666666">-=</span> eta <span style="color: #666666">*</span> dBo
hidden_weights <span style="color: #666666">-=</span> eta <span style="color: #666666">*</span> dWh
hidden_bias <span style="color: #666666">-=</span> eta <span style="color: #666666">*</span> dBh
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;New accuracy on training data: &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(accuracy_score(predict(X_train), Y_train)))
</pre></div>
<p>
<p>
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