深度学习(1)Python手写Numpy神经网络

对神经网络实现时,为了追求效率一般采用向量化技术,而尽可能的减少显式使用for循环的次数。所谓向量化,就是采用矩阵运算来代替循环,将上一篇文章《深度学习(0)Logistics正向、反向传播推导》,向量化如下,

  • \(W=\left[w_{1}, w_{2} \dots w_{m}\right]\)
  • \(X=\left[x^{(1)}, x^{(2)} \dots x^{(m)}\right]\)
  • \(B=\left[b, b, \dots b\right]\)
  • \(Z=\left[z^{(1)}, z^{(2)} \dots z^{(m)}\right]\)
  • \(A=\left[a^{(1)}, a^{(2)} \dots a^{(m)}\right]\)
  • \(Y=\left[y^{(1)}, y^{(2)} \dots y^{(m)}\right]\)
  • \(dW=\left[dw_{1}, dw_{2} \dots dw_{m}\right]^T\)

其中,单个样本\(x_i\)的维度为n,每一批输入样本个数为m。

反向传播

上一篇文章《深度学习(0)Logistics正向、反向传播推导》已经对\(dW\)\(dB\)进行了详细推导, \[\frac{d J}{d w_{j}}=\frac{1}{m} \sum_{i=1}^{m} \frac{d L^{(i)}}{d a^{(i)}} \frac{d a^{(i)}}{d z^{(i)}} \frac{d z^{(i)}}{d w_{j}} = \frac{1}{m} \sum_{i=1}^{m}\left(a^{(i)}-y^{(i)}\right) x_{j}^{(i)}\]

\[\frac{d J}{d b}=\frac{1}{m} \sum_{i=1}^{m} \frac{d L^{(i)}}{d a^{(i)}} \frac{d a^{(i)}}{d z^{(i)}} \frac{d z^{(i)}}{d b} = \frac{1}{m} \sum_{i=1}^{m}\left(a^{(i)}-y^{(i)}\right)\]

这里从向量化角度再简单推导一下,矩阵转置,\(Z^T = (W^TX+B)^T=X^TW+B^T\),两边对\(W\)求导得\(dZ^T=X^TdW\),再左右同时“左乘”\(X\)于是有,\(dW = \frac{1}{m} X dZ^T\)。注意\(dZ\)就是变化的微分,也就是\(dZ=A-Y\),于是有,

import random
import numpy as np

class Network(object):

def __init__(self, sizes):
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):
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):
if test_data:
test_data = list(test_data)
n_test = len(test_data)
training_data = list(training_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 %d: %d / %d"%(j, self.evaluate(test_data), n_test))
else:
print("Epoch %d complete"%j)

def update_mini_batch(self, mini_batch, eta):
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):
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())
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):
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)

#### Miscellaneous functions
def sigmoid(z):
return 1.0/(1.0+np.exp(-z))

def sigmoid_prime(z):
return sigmoid(z)*(1-sigmoid(z))

Reference:

Neural Networks and Deep Learning