随机梯度下降随机梯度下降SGD算法原理和实现算法原理和实现
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:
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