# %load network.py """ network.py ~~~~~~~~~~ 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(object): def __init__(self, sizes): """ The list "size" 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(np.dot(w,a)+b) return a def SGD(self, training_data, epochs, mini_batch_size, eta, 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. If "test_data" is provided then the network will be evaluated against the test data after each epoch, and partial progress printed out. This is useful for tracking progress, but slows things down substantially. @training_data: a list of tuples(x, y) representing the training inputs and corresponding desired outputs. @epochs: the number of epochs to train for. @mini_batch_size: the size of the mini_batches to use when sampling. @eta: learning rate $\eta$ @test_data: if supplied, the program will evaluate the network after each epoch of training, and print out partial progress. """ training_data = list(training_data) n = len(training_data) if test_data: test_data = list(test_data) n_test = len(test_data) for j in range(epochs): random.shuffle(training_data) mini_batches = [ training_data[k:k+mini_batch_size] for k in range(0, n, mini_batch_size) ] for mini_batch in mini_batches: self.update_mini_batch(mini_batch, eta) if test_data: print("Epoch {}: {} / {}".format(j, self.evaluate(test_data), n_test)); else: print("Epoch {} complete".format(j)) def update_mini_batch(self, mini_batch, eta): """ Update the network's weights and biases by applying gradient descent using backpropagation to a single mini batch. The "mini_batch" is a list of tuples "(x, y)", and "eta" is the learning rate. """ nabla_b = [np.zeros(b.shape) for b in self.biases] nabla_w = [np.zeros(w.shape) for w in self.weights] for x,y in mini_batch: delta_nabla_b, delta_nabla_w = self.backprop(x,y) nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)] nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)] self.weights = [w-(eta/len(mini_batch))*nw for w, nw in zip(self.weights, nabla_w)] self.biases = [b-(eta/len(mini_batch))*nb for b, nb in zip(self.biases, nabla_b)] def backprop(self, x, y): """Return a tuple "(nabla_b, nabla_w)" representing the gradient for the cost function C_x. "nabla_b" and "nabla_w" are layer-by-layer lists of numpy arrays, similar to "self.biases" and "self.weights".""" nabla_b = [np.zeros(b.shape) for b in self.biases] nabla_w = [np.zeros(w.shape) for w in self.weights] # feedforward activation = x activations = [x] # list to store all the activations, layer by layer zs = [] # list to store all the z vectors, layer by layer for b,w in zip(self.biases, self.weights): z = np.dot(w, activation) + b zs.append(z) activation = sigmoid(z) activations.append(activation) # backward pass delta = self.cost_derivative(activations[-1],y)*sigmoid_prime(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 notation in Chapter 2 of 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 in the book, used here to take advantage of the fact that Python can use negative indices in lists. """ for l in range(2, self.num_layers): z = zs[-l] sp = sigmoid_prime(z) delta = np.dot(self.weights[-l+1].transpose(), delta) * sp nabla_b[-l] = delta nabla_w[-l] = np.dot(delta, activations[-l-1].transpose()) return (nabla_b, nabla_w) def evaluate(self, test_data): """Return the number of test inputs for which the neural network outputs the correct result. 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)),y) for (x,y) in test_data] return sum(int(x==y) for (x,y) in test_results) def cost_derivative(self, output_activations, y): """Return the vector of partial derivatives \partial C_x / \partial a for the output activations.""" return (output_activations - y) #### Miscellaneous functions def sigmoid(z): """The sigmoid function.""" return 1.0/(1.0 + np.exp(-z)) def sigmoid_prime(z): """Derivative of the sigmoid function.""" return sigmoid(z)*(1-sigmoid(z))