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# 《神经网络和深度学习》系列文章十六：反向传播算法代码

## 目录

1、使用神经网络识别手写数字

2、反向传播算法是如何工作的

• 热身：一个基于矩阵的快速计算神经网络输出的方法
• 关于损失函数的两个假设
• 反向传播背后的四个基本等式
• 四个基本等式的证明（选读）
• 反向传播算法
• 反向传播算法代码
• 什么时候反向传播算法高效
• 反向传播算法再理解

3、改进神经网络的学习方法

4、神经网络能够计算任意函数的视觉证明

5、为什么深度神经网络的训练是困难的

6、深度学习

class Network(object):
...
def update_mini_batch(self, mini_batch, eta):
"""Update the network's weights and biases by applying
gradient 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]
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)]

class Network(object):
...
def backprop(self, x, y):
"""Return a tuple "(nabla_b, nabla_w)" representing the
gradient for the cost function C_x.  "nabla_b" and
"nabla_w" are layer-by-layer lists of numpy arrays, similar
to "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]
# 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())
# 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) * sp
nabla_b[-l] = delta
nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
return (nabla_b, nabla_w)

...

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)

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))

### 问题

• 在一个批次（微型批次）上应用完全基于矩阵的反向传播方法