Adding network_basic.py

Author: mnielsen

code/network_basic.py

@@ -0,0 +1,162 @@
+"""
+ network_basic
+~~~~~~~~~~~~~~
+
+A module to implement the stochastic gradient descent learning
+algorithm for a feedforward neural network.  Gradients are calculated
+using backpropagation.  Note that I have focused on making the code
+simple, easily readable, and easily modifiable.  It is not optimized,
+and omits many desirable features.
+"""
+
+#### Libraries
+# Standard library
+import random
+
+# Third-party libraries
+import numpy as np
+
+class Network():
+
+    def __init__(self, sizes):
+        """The list ``sizes`` contains the number of neurons in the
+        respective layers of the network.  For example, if the list
+        was [2, 3, 1] then it would be a three-layer network, with the
+        first layer containing 2 neurons, the second layer 3 neurons,
+        and the third layer 1 neuron.  The biases and weights for the
+        network are initialized randomly, using a Gaussian
+        distribution with mean 0, and variance 1.  Note that the first
+        layer is assumed to be an input layer, and by convention we
+        won't set any biases for those neurons, since biases are only
+        ever used in computing the outputs from later layers."""
+        self.num_layers = len(sizes)
+        self.sizes = sizes
+        self.biases = [np.random.randn(y, 1) for y in sizes[1:]]
+        self.weights = [np.random.randn(y, x) 
+                        for x, y in zip(sizes[:-1], sizes[1:])]
+
+    def feedforward(self, a):
+        "Return the output of the network if ``a`` is input."
+        for b, w in zip(self.biases, self.weights):
+            a = sigmoid_vec(np.dot(w, a)+b)
+        return a
+
+    def SGD(self, training_data, epochs, mini_batch_size, eta,
+            lmbda, test=False, test_data=None)
+        """Train the neural network using mini-batch stochastic
+        gradient descent.  The ``training_data`` is a list of tuples
+        ``(x, y)`` representing the training inputs and the desired
+        outputs.  The other non-optional parameters are
+        self-explanatory.  Set ``test`` to ``True`` to evaluate the
+        network against the test data after each epoch, and to print
+        out partial progress.  This is useful for tracking progress,
+        but slows things down substantially.  If ``test`` is set, then
+        appropriate ``test_data`` must be supplied.
+        """
+        if test: n_test = len(test_inputs)
+        n = len(training_data)
+        for j in xrange(epochs):
+            random.shuffle(training_data)
+            mini_batches = [
+                training_data[k:k+mini_batch_size]
+                for k in xrange(0, len(training_data), mini_batch_size)]
+            for mini_batch in mini_batches:
+                self.backprop(mini_batch, n, eta, lmbda)
+            if test:
+                print "Epoch {}: {} / {}".format(
+                    j, self.evaluate(test_inputs, actual_test_results), n_test)
+            else:
+                print "Epoch %s complete" % j
+
+    def backprop(self, training_data, T, eta, lmbda):
+        """Update the network's weights and biases by applying a
+        single iteration of gradient descent using backpropagation.
+        The ``training_data`` is a list of tuples ``(x, y)``.  It need
+        not include the entire training data set --- it might be a
+        mini-batch, or even a single training example.  ``T`` is the
+        size of the total training set (which may not be the same as
+        the size of ``training_data``).  The other parameters are
+        self-explanatory."""
+        nabla_b = [np.zeros(b.shape) for b in self.biases]
+        nabla_w = [np.zeros(w.shape) for w in self.weights]
+        B = len(training_data)
+        for x, y in training_data:
+            # feedforward
+            activation = x
+            activations = [x] # list to store all the activations
+            zs = [] # list to store all the z vectors
+            for b, w in zip(self.biases, self.weights):
+                z = np.dot(w, activation)+b
+                zs.append(z)
+                activation = sigmoid_vec(z)
+                activations.append(activation)
+            # backward pass
+            delta = self.cost_derivative(activations[-1], y) * \
+                sigmoid_prime_vec(zs[-1])
+            nabla_b[-1] += delta
+            nabla_w[-1] += np.dot(delta, activations[-2].transpose())
+            # Note that the variable l in the loop below is used a
+            # little differently to the book.  Here, l = 1 means the
+            # last layer of neurons, l = 2 is the second-last layer,
+            # and so on.  It's a renumbering of the scheme used in the
+            # book, used to take advantage of the fact that Python can
+            # use negative indices in lists.
+            for l in xrange(2, self.num_layers):
+                z = zs[-l]
+                spv = sigmoid_prime_vec(z)
+                delta = np.dot(self.weights[-l+1].transpose(), delta) * spv
+                nabla_b[-l] += delta
+                nabla_w[-l] += np.dot(delta, activations[-l-1].transpose())
+        # Add the regularization terms to the gradient for the weights
+        nabla_w = [nw+(lmbda*B/T)*w for nw, w in zip(nabla_w, self.weights)]
+        self.weights = [w-eta*nw for w, nw in zip(self.weights, nabla_w)]
+        self.biases = [b-eta*nb for b, nb in zip(self.biases, nabla_b)]
+
+    def evaluate(self, test_inputs, actual_test_results):
+        """Return the number of ``test_inputs`` for which the neural
+        network outputs the correct result, i.e., the same result as
+        given in ``actual_test_results``.  Note that the neural
+        network's output is assumed to be the index of whichever
+        neuron in the final layer has the highest activation."""
+        test_results = [np.argmax(self.feedforward(x)) for x in test_inputs]
+        return sum(int(x == y) 
+                   for x, y in zip(test_results, actual_test_results))
+        
+    def cost(self, x, y):
+        """Return the quadratic cost associated to the network, with
+        input ``x`` and desired output ``y``.  Note that there is no
+        regularization."""
+        return np.sum((self.feedforward(x)-y)**2)/2.0
+
+    def cost_derivative(self, output_activations, y):
+        """Return the vector of partial derivatives \partial C_x /
+        \partial a for the output activations, ``a``.  For the
+        unregularized quadratic cost this is just the difference
+        between the output activations and the desired output, ``y``."""
+        return (output_activations-y) 
+
+    def evaluate_training_results(self, training_data):
+        """Return the number of elements of the ``training_data`` that
+        are correctly classified."""
+        training_results = [np.argmax(self.feedforward(x[0])) for x in 
+                            training_data]
+        actual_training_results = [np.argmax(x[1]) for x in training_data]
+        return sum(int(x == y) 
+                   for x, y in zip(training_results, actual_training_results))
+
+#### Miscellaneous functions
+def sigmoid(z):
+    """The sigmoid function.  Note that it checks to see whether ``z``
+    is very negative, to avoid overflow errors in the exponential
+    function.  No corresponding test of ``z`` being very positive is
+    necessary --- ordinary Python arithmetic deals just fine with that
+    case."""
+    return 0.0 if z < -700 else 1.0/(1.0+np.exp(-z))
+
+sigmoid_vec = np.vectorize(sigmoid)
+
+def sigmoid_prime(z):
+    """Derivative of the sigmoid function."""
+    return sigmoid(z)*(1-sigmoid(z))
+
+sigmoid_prime_vec = np.vectorize(sigmoid_prime)