- 神经网络从原理到实现

有两点需要强调下：
  •   （2）中的结果比（1）中的结果多了一个求和公式，这是因为计算隐层与输入层之间的参数时，输出层与隐层的每一个节点都有影响。
  •   （2）中参数更新的结果可以复用（1）中的参数更新结果，从某种程度上，与反向传播这个算法名称不谋而合，不得不惊叹。

def backprop(self, x, y): """返回一个元组(nabla_b, nabla_w)代表目标函数的梯度.""" 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()) """l = 1 表示最后一层神经元，l = 2 是倒数第二层神经元, 依此类推.""" for l in xrange(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)

4. 完整代码实现

# -*- coding: utf-8 -*- import random import numpy as np class Network(object): def __init__(self, sizes): """参数sizes表示每一层神经元的个数，如[2,3,1],表示第一层有2个神经元，第二层有3个神经元，第三层有1个神经元.""" 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): """前向传播""" 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): """随机梯度下降""" if test_data: n_test = len(test_data) 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, n, mini_batch_size)] 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): """使用后向传播算法进行参数更新.mini_batch是一个元组(x, y)的列表、eta是学习速率""" 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): """返回一个元组(nabla_b, nabla_w)代表目标函数的梯度.""" nabla_b = [np.zeros(b.shape) for b in self.biases] nabla_w = [np.zeros(w.shape) for w in self.weights] # 前向传播 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()) """l = 1 表示最后一层神经元，l = 2 是倒数第二层神经元, 依此类推.""" for l in xrange(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): """返回分类正确的个数""" 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 (output_activations-y) def sigmoid(z): return 1.0/(1.0+np.exp(-z)) def sigmoid_prime(z): """sigmoid函数的导数""" return sigmoid(z)*(1-sigmoid(z))

5. 简单应用

# -*- coding: utf-8 -*- from network import * def vectorized_result(j,nclass): """离散数据进行one-hot""" e = np.zeros((nclass, 1)) e[j] = 1.0 return e def get_format_data(X,y,isTest): ndim = X.shape[1] nclass = len(np.unique(y)) inputs = [np.reshape(x, (ndim, 1)) for x in X] if not isTest: results = [vectorized_result(y,nclass) for y in y] else: results = y data = zip(inputs, results) return data #随机生成数据 from sklearn.datasets import * np.random.seed(0) X, y = make_moons(200, noise=0.20) ndim = X.shape[1] nclass = len(np.unique(y)) #划分训练、测试集 from sklearn.cross_validation import train_test_split train_x,test_x,train_y,test_y = train_test_split(X,y,test_size=0.2,random_state=0) training_data = get_format_data(train_x,train_y,False) test_data = get_format_data(test_x,test_y,True) net = Network(sizes=[ndim,10,nclass]) net.SGD(training_data=training_data,epochs=5,mini_batch_size=10,eta=0.1,test_data=test_data)

参考文献
（1）周志华《机器学习》
（2）https://github.com/mnielsen/neural-networks-and-deep-learning
（3）https://zhuanlan.zhihu.com/p/21525237

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原文链接：https://blog.csdn.net/a819825294/article/details/53393837