# -*- coding: utf-8 -*- ''' Created on 2018年4月19日 @author: userhttps://github.com/mnielsen/neural-networks-and-deep-learning @summary: MLP(MultiLayer Perceptrons) 多層神經網絡 sigmoid neurons 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 import time import mnist_loader from PIL import Image # Third-party libraries import numpy as npclass Network(object):def __init__(self, sizes):"""The list ``sizes`` contains the number of neurons in therespective layers of the network. For example, if the listwas [2, 3, 1] then it would be a three-layer network, with thefirst layer containing 2 neurons, the second layer 3 neurons,and the third layer 1 neuron. The biases and weights for thenetwork are initialized randomly, using a Gaussiandistribution with mean 0, and variance 1. Note that the firstlayer is assumed to be an input layer, and by convention wewon't set any biases for those neurons, since biases are onlyever used in computing the outputs from later layers."""self.num_layers = len(sizes)self.sizes = sizes#隨機初始化偏置和權重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:])]#前向傳播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 adef SGD(self, training_data, epochs, mini_batch_size, eta,test_data=None):"""Train the neural network using mini-batch stochasticgradient descent. The ``training_data`` is a list of tuples``(x, y)`` representing the training inputs and the desiredoutputs. The other non-optional parameters areself-explanatory. If ``test_data`` is provided then thenetwork will be evaluated against the test data after eachepoch, and partial progress printed out. This is useful fortracking progress, but slows things down substantially."""if test_data: n_test = len(test_data)n = len(training_data)for j in xrange(epochs):random.shuffle(training_data)#mini_batches是訓練數據分批mini_batches = [training_data[k:k+mini_batch_size]for k in xrange(0, n, mini_batch_size)]#每個mini_batch都更新一次,重復完整個數據集for mini_batch in mini_batches:self.update_mini_batch(mini_batch, eta)if test_data:print "Epoch {0}: {1} / {2}".format(j, self.evaluate(test_data), n_test)else:print "Epoch {0} complete".format(j)def update_mini_batch(self, mini_batch, eta):"""Update the network's weights and biases by applyinggradient 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]#mini_batch中的一個實例調用梯度下降得到各個參數的偏導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))*nwfor w, nw in zip(self.weights, nabla_w)]self.biases = [b-(eta/len(mini_batch))*nbfor b, nb in zip(self.biases, nabla_b)]def backprop(self, x, y):"""Return a tuple ``(nabla_b, nabla_w)`` representing thegradient for the cost function C_x. ``nabla_b`` and``nabla_w`` are layer-by-layer lists of numpy arrays, similarto ``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]# feedforwardactivation = xactivations = [x] # list to store all the activations, layer by layer#存儲所有的激活值,一層一層的形式zs = [] # list to store all the z vectors, layer by layerfor b, w in zip(self.biases, self.weights):z = np.dot(w, activation)+bzs.append(z)activation = sigmoid(z)activations.append(activation)# backward passdelta = self.cost_derivative(activations[-1], y) * \sigmoid_prime(zs[-1])nabla_b[-1] = deltanabla_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 xrange(2, self.num_layers):z = zs[-l]sp = sigmoid_prime(z)delta = np.dot(self.weights[-l+1].transpose(), delta) * spnabla_b[-l] = deltanabla_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 neuralnetwork outputs the correct result. Note that the neuralnetwork's output is assumed to be the index of whicheverneuron 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))if __name__ == "__main__": start = time.clock() training_data,validation_data,test_data = mnist_loader.load_data_wrapper()#顯示圖片I = training_data[0][0]I.resize((28, 28))im = Image.fromarray((I*256).astype('uint8'))im.show()#MLP訓練net = Network([784,30,10])net.SGD(training_data,30,10,3.0,test_data=test_data)end = time.clock() print('finish all in %s' % str(end - start))
# -*- coding: utf-8 -*- ''' Created on 2018年4月19日@author: user ''' """ mnist_loader ~~~~~~~~~~~~ A library to load the MNIST image data. For details of the data structures that are returned, see the doc strings for ``load_data`` and ``load_data_wrapper``. In practice, ``load_data_wrapper`` is the function usually called by our neural network code. """#### Libraries # Standard library import cPickle import gzip# Third-party libraries import numpy as npdef load_data():"""Return the MNIST data as a tuple containing the training data,the validation data, and the test data.The ``training_data`` is returned as a tuple with two entries.The first entry contains the actual training images. This is anumpy ndarray with 50,000 entries. Each entry is, in turn, anumpy ndarray with 784 values, representing the 28 * 28 = 784pixels in a single MNIST image.The second entry in the ``training_data`` tuple is a numpy ndarraycontaining 50,000 entries. Those entries are just the digitvalues (0...9) for the corresponding images contained in the firstentry of the tuple.The ``validation_data`` and ``test_data`` are similar, excepteach contains only 10,000 images.This is a nice data format, but for use in neural networks it'shelpful to modify the format of the ``training_data`` a little.That's done in the wrapper function ``load_data_wrapper()``, seebelow."""f = gzip.open('./data/mnist.pkl.gz', 'rb')training_data, validation_data, test_data = cPickle.load(f)f.close()return (training_data, validation_data, test_data)def load_data_wrapper():"""Return a tuple containing ``(training_data, validation_data,test_data)``. Based on ``load_data``, but the format is moreconvenient for use in our implementation of neural networks.In particular, ``training_data`` is a list containing 50,0002-tuples ``(x, y)``. ``x`` is a 784-dimensional numpy.ndarraycontaining the input image. ``y`` is a 10-dimensionalnumpy.ndarray representing the unit vector corresponding to thecorrect digit for ``x``.``validation_data`` and ``test_data`` are lists containing 10,0002-tuples ``(x, y)``. In each case, ``x`` is a 784-dimensionalnumpy.ndarry containing the input image, and ``y`` is thecorresponding classification, i.e., the digit values (integers)corresponding to ``x``.Obviously, this means we're using slightly different formats forthe training data and the validation / test data. These formatsturn out to be the most convenient for use in our neural networkcode."""tr_d, va_d, te_d = load_data()training_inputs = [np.reshape(x, (784, 1)) for x in tr_d[0]]training_results = [vectorized_result(y) for y in tr_d[1]]training_data = zip(training_inputs, training_results)validation_inputs = [np.reshape(x, (784, 1)) for x in va_d[0]]validation_data = zip(validation_inputs, va_d[1])test_inputs = [np.reshape(x, (784, 1)) for x in te_d[0]]test_data = zip(test_inputs, te_d[1])return (training_data, validation_data, test_data)def vectorized_result(j):"""Return a 10-dimensional unit vector with a 1.0 in the jthposition and zeroes elsewhere. This is used to convert a digit(0...9) into a corresponding desired output from the neuralnetwork."""e = np.zeros((10, 1))e[j] = 1.0return e

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