BP神经网络结构与原理
参数表示
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$n_l$ ：表示网络层数，此处为 4
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$L_l$ ：表示第l层， L1 是输入层，$L_n$ 是输出层，其他为隐含层
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$w^{(l)}_{ij}$ ：表示第l +1层第i 个单元与第l 层第 j 个单元的连接权重
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$b^{(l)}_i$ ：表示第l 层第i 个单元的偏置项(激活阈值)
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$z^{(l)}_i$ ：表示第l 层第i 个单元的权重累计
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$a^{(l)}_i$ ：表示第l 层第i 个单元的激活值(输出值) (sigmod函数映射得到$f(z_i) = a_i$)
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$h_{w,b}(X)$ ：表示最后的输出值
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$S_l$ ：表示第l 层的神经元个数
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样本个数为m ，特征个数为 n
参数间的关系
BP网络完整流程
BP神经网络和普通神经网络的区别（个人理解）:
普通神经网络在进行前向传播更新参数时是通过链式法则求导得到（如果有多层神经网络的话，链将会变得特别长），而BP神经网络是通过对两层神经网络（如最后一列和倒数第一列）进行前向传播求导得到一个通式，然后如果求倒数第二列和倒数第三列的话只需将上一个通式中的参数进行改变
BP神经网络第一种实现
采用误差平方和作为损失函数，基于反向传播算法推导，可得最终的 4 个方程式，如下：
前向计算
反向传播
BP神经网络的改进和第二种实现
BP神经网络的改进
交叉熵代价函数
“严重错误”导致学习缓慢，如果在初始化权重和偏置时，故意产生一个背离预期较大的输出，那么训练网络的过程中需要用很多次迭代，才能抵消掉这种背离，恢复正常的学习。
交叉熵可以理解为代替了误差平方和
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引入交叉熵代价函数目的是解决一些实例在刚开始训练时学习得非常慢的问题，其主要针对激活函数为 Sigmod 函数
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如果采用一种不会出现饱和状态的激活函数（如ReLu），那么可以继续使用误差平方和作为损失函数
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如果在输出神经元是 S 型神经元时，交叉熵一般都是更好的选择
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输出神经元是线性的那么二次代价函数不再会导致学习速度下降的问题。在此情形下，二次代价函数就是一种合适的选择
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交叉熵无法改善隐藏层中神经元发生的学习缓慢
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交叉熵损失函数只对网络输出“明显背离预期”时发生的学习缓慢有改善效果
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应用交叉熵损失并不能改善或避免神经元饱和，而是当输出层神经元发生饱和时， 能够避免其学习缓慢的问题
第二种实现
损失函数采用交叉熵代价函数
权重初始化时，偏差初始化方法不变，将原始的方差除以一个np.sqrt(x)，减少方差，避免饱和
代码实现;
#### Libraries
# Standard library
import json
import random
import sys
# Third-party libraries
import numpy as np
#### Define the quadratic and cross-entropy cost functions
import mnist_loader
class QuadraticCost(object): # 误差平方和代价函数
@staticmethod
def fn(a, y):
return 0.5*np.linalg.norm(a-y)**2
@staticmethod
def delta(z, a, y):
"""Return the error delta from the output layer."""
return (a-y) * sigmoid_prime(z)
class CrossEntropyCost(object): # 交叉熵代价函数
@staticmethod
def fn(a, y):
return np.sum(np.nan_to_num(-y*np.log(a)-(1-y)*np.log(1-a))) # 使用0代替nan 一个较大值代替inf
@staticmethod
def delta(z, a, y):
return (a-y)
#### Main Network class
class Network(object):
def __init__(self, sizes, cost=CrossEntropyCost): #采用交叉熵代价函数
self.num_layers = len(sizes)
self.sizes = sizes
self.default_weight_initializer()
self.cost=cost
def default_weight_initializer(self): # 推荐的权重初始化方式
self.biases = [np.random.randn(y, 1) for y in self.sizes[1:]] # 偏差初始化方式不变
self.weights = [np.random.randn(y, x)/np.sqrt(x) # 将方差减少，避免饱和
for x, y in zip(self.sizes[:-1], self.sizes[1:])]
def large_weight_initializer(self): # 不推荐的初始化方式
self.biases = [np.random.randn(y, 1) for y in self.sizes[1:]]
self.weights = [np.random.randn(y, x)
for x, y in zip(self.sizes[:-1], self.sizes[1:])]
def feedforward(self, a):
"""Return the output of the network if ``a`` is input."""
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,
lmbda = 0.0,
evaluation_data=None,
monitor_evaluation_cost=False,
monitor_evaluation_accuracy=False,
monitor_training_cost=False,
monitor_training_accuracy=False,
early_stopping_n = 0):
# early stopping functionality:
best_accuracy=1
training_data = list(training_data)
n = len(training_data)
if evaluation_data:
eval（evaluation_data)
n_data = len(evaluation_data)
# early stopping functionality:
best_accuracy=0
no_accuracy_change=0
evaluation_cost, evaluation_accuracy = [], []
training_cost, training_accuracy = [], []
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, lmbda, len(training_data))
print("Epoch %s training complete" % j)
if monitor_training_cost:
cost = self.total_cost(training_data, lmbda)
training_cost.append(cost)
print("Cost on training data: {}".format(cost))
if monitor_training_accuracy:
accuracy = self.accuracy(training_data, convert=True)
training_accuracy.append(accuracy)
print("Accuracy on training data: {} / {}".format(accuracy, n))
if monitor_evaluation_cost:
cost = self.total_cost(evaluation_data, lmbda, convert=True)
eval（cost)
print("Cost on eval（cost))
if monitor_evaluation_accuracy:
accuracy = self.accuracy(evaluation_data)
eval（accuracy)
print("Accuracy on eval（self.accuracy(evaluation_data), n_data))
# Early stopping:
if early_stopping_n > 0: # 如果采用了早停止策略，则当准确率超过设定的次数依然没有改变，则停止训练
if accuracy > best_accuracy:
best_accuracy = accuracy
no_accuracy_change = 0
print("Early-stopping: Best so far {}".format(best_accuracy))
else:
no_accuracy_change += 1
if (no_accuracy_change == early_stopping_n):
print("Early-stopping: No accuracy change in last epochs: {}".format(early_stopping_n))
return evaluation_cost, evaluation_accuracy, training_cost, training_accuracy
return evaluation_cost, evaluation_accuracy, \
training_cost, training_accuracy
def update_mini_batch(self, mini_batch, eta, lmbda, n):
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 = [(1-eta*(lmbda/n))*w-(eta/len(mini_batch))*nw # 带L2范数的权重更新
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).delta(zs[-1], activations[-1], y) # 这里和以前不一样，其他地方一样
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 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 accuracy(self, data, convert=False):
if convert: # 训练集使用-由于训练集的输出是独热码，而其他数据不是
results = [(np.argmax(self.feedforward(x)), np.argmax(y))
for (x, y) in data]
else: # 验证集合测试集使用
results = [(np.argmax(self.feedforward(x)), y)
for (x, y) in data]
result_accuracy = sum(int(x == y) for (x, y) in results)
return result_accuracy
def total_cost(self, data, lmbda, convert=False):
cost = 0.0
for x, y in data:
a = self.feedforward(x)
if convert: y = vectorized_result(y) # 测试集和验证集需要向量化
cost += self.cost.fn(a, y)/len(data) # 带L2范数的代价函数
cost += 0.5*(lmbda/len(data))*sum(np.linalg.norm(w)**2 for w in self.weights) # '**' - to the power of.
return cost
def save(self, filename):
data = {"sizes": self.sizes,
"weights": [w.tolist() for w in self.weights],
"biases": [b.tolist() for b in self.biases],
"cost": str(self.cost.__name__)}
f = open(filename, "w")
json.dump(data, f)
f.close()
#### Loading a Network
def load(filename):
f = open(filename, "r")
data = json.load(f)
f.close()
cost = getattr(sys.modules[__name__], data["cost"])
net = Network(data["sizes"], cost=cost)
net.weights = [np.array(w) for w in data["weights"]]
net.biases = [np.array(b) for b in data["biases"]]
return net
#### Miscellaneous functions
def vectorized_result(j):
e = np.zeros((10, 1))
e[j] = 1.0
return e
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))
if __name__ == '__main__':
training_data, validation_data, test_data = mnist_loader.load_data_wrapper()
training_data = list(training_data)
net = Network([784, 30, 10], cost=CrossEntropyCost)
# net.large_weight_initializer()
net.SGD(training_data, 30, 10, 0.1, lmbda=5.0, evaluation_data=validation_data,
monitor_evaluation_accuracy=True)
运行结果：
