随机梯度下降SGD算法原理和实现

backpropagation

backpropagation解决的核心问题损失函数c与w,b求偏导，(c为cost(w,b))

整体来说，分两步

1.z=w*a’+b
2.a=sigmoid(z)
其中，a’表示上一层的输出值，a表示当前该层的输出值
1，输入x，正向的更新一遍所有的a值就都有了，
2，计算输出层的delta=(y-a)点乘sigmoid(z)函数对z的偏导数
3，计算输出层之前层的误差delta，该delta即为损失函数对b的偏导数，
4，然后根据公式4，求出对w的偏导数
公式推导详解

import numpy as np
import random

class Network(object):
    def __init__(self, sizes):
        self.number_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):
        for b, w in zip(self.biases, self.weights):
            a = sigmoid(np.dot(w, a) + b)
        return a
    def evaluate(self,test_data):
        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 derivate(self,output,y):
        return (output-y)
    def backprop(self,x,y):
        nabla_b = [np.zeros(b.shape) for b in self.biases]
        nabla_w = [np.zeros(w.shape) for w in self.weights]
        activation = x
        activations = [x]
        zs = []
        for b, w in zip(self.biases, self.weights):
            z = np.dot(w, activation)+b
            zs.append(z)
            activation = sigmoid(z)
            activations.append(activation)
        delta = self.derivate(activations[-1], y) * sigmoid_prime(zs[-1])
        nabla_b[-1] = delta
        nabla_w[-1] = np.dot(delta, activations[-2].transpose())
        for i in range(2,self.number_layers):
            z = zs[-i]
            ps = sigmoid_prime(z)
            delta = np.dot(self.weights[-i+1].transpose(), delta) * ps
            nabla_b[-i] = delta
            nabla_w[-i] = np.dot(delta, activations[-i-1].transpose())
        return nabla_b, nabla_w
    def update_mini_batch(self, mini_batch, eta):
        nabla_w = [np.zeros(w.shape) for w in self.weights]
        nabla_b = [np.zeros(b.shape) for b in self.biases]
        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 SGD(self, training_data, epochs, mini_batch_size, eta, test_data=None):
        if test_data:n_test = len(test_data)
        n = len(training_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{0} : {1}/{2} '.format(j, self.evaluate(test_data), n_test))
            else:
                print('Epoch complete!'.format(j))


def sigmoid(z):
        return (1.0 / (1.0+np.exp(-z)))
def sigmoid_prime(z):
        return sigmoid(z) * (1-sigmoid(z))