BP神经网络——交叉熵作代价函数

Sigmoid函数

当神经元的输出接近 1时，曲线变得相当平，即σ′(z)的值会很小，进而也就使?C/?w和?C/?b会非常小。造成学习缓慢，下面有一个二次代价函数的cost变化图，epoch从15到50变化很小。

引入交叉熵代价函数

针对上述问题，希望对输出层选择一个不包含sigmoid的权值更新，使得

由链式法则，得到

由σ′(z) = σ(z)(1? σ(z))以及σ(z)=a，可以将上式转换成

对方程进行关于a的积分，可得

对样本进行平均之后就是下面的交叉熵代价函数

对比之前的输出层delta，相当于去掉了前面的

相应的代码仅改动了一行(58->59)，新的cost变化图如下。

在训练和测试数据各5000个时，识别正确数从4347稍提高到4476。

 1 # coding:utf8
 2 import cPickle
 3 import numpy as np
 4 import matplotlib.pyplot as plt
 5
 6
 7 class Network(object):
 8     def __init__(self, sizes):
 9         self.num_layers = len(sizes)
10         self.sizes = sizes
11         self.biases = [np.random.randn(y, 1) for y in sizes[1:]]  # L(n-1)->L(n)
12         self.weights = [np.random.randn(y, x)
13                         for x, y in zip(sizes[:-1], sizes[1:])]
14
15     def feedforward(self, a):
16         for b_, w_ in zip(self.biases, self.weights):
17             a = self.sigmoid(np.dot(w_, a)+b_)
18         return a
19
20     def SGD(self, training_data, test_data,epochs, mini_batch_size, eta):
21         n_test = len(test_data)
22         n = len(training_data)
23         plt.xlabel(‘epoch‘)
24         plt.ylabel(‘Accuracy‘)
25         plt.title(‘cost‘)
26         cy=[]
27         cx=range(epochs)
28         for j in cx:
29             self.cost = 0.0
30             np.random.shuffle(training_data)  # shuffle
31             for k in xrange(0, n, mini_batch_size):
32                 mini_batch = training_data[k:k+mini_batch_size]
33                 self.update_mini_batch(mini_batch, eta)
34             cy.append(self.cost/n)
35             print "Epoch {0}: {1} / {2}".format(
36                     j, self.evaluate(test_data), n_test)
37         plt.plot(cx,cy)
38         plt.scatter(cx,cy)
39         plt.show()
40
41     def update_mini_batch(self, mini_batch, eta):
42         for x, y in mini_batch:
43             delta_b, delta_w,cost = self.backprop(x, y)
44             self.weights -= eta/len(mini_batch)*delta_w
45             self.biases -= eta/len(mini_batch)*delta_b
46             self.cost += cost
47
48     def backprop(self, x, y):
49         b=np.zeros_like(self.biases)
50         w=np.zeros_like(self.weights)
51         a_ = x
52         a = [x]
53         for b_, w_ in zip(self.biases, self.weights):
54             a_ = self.sigmoid(np.dot(w_, a_)+b_)
55             a.append(a_)
56         for l in xrange(1, self.num_layers):
57             if l==1:
58                 # delta= self.sigmoid_prime(a[-1])*(a[-1]-y)  # O(k)=a[-1], t(k)=y
59                 delta= a[-1]-y  # cross-entropy
60             else:
61                 sp = self.sigmoid_prime(a[-l])   # O(j)=a[-l]
62                 delta = np.dot(self.weights[-l+1].T, delta) * sp
63             b[-l] = delta
64             w[-l] = np.dot(delta, a[-l-1].T)
65         cost=0.5*np.sum((a[-1]-y)**2)
66         return (b, w,cost)
67
68     def evaluate(self, test_data):
69         test_results = [(np.argmax(self.feedforward(x)), y)
70                         for (x, y) in test_data]
71         return sum(int(x == y) for (x, y) in test_results)
72
73     def sigmoid(self,z):
74         return 1.0/(1.0+np.exp(-z))
75
76     def sigmoid_prime(self,z):
77         return z*(1-z)
78
79 if __name__ == ‘__main__‘:
80
81         def get_label(i):
82             c=np.zeros((10,1))
83             c[i]=1
84             return c
85
86         def get_data(data):
87             return [np.reshape(x, (784,1)) for x in data[0]]
88
89         f = open(‘mnist.pkl‘, ‘rb‘)
90         training_data, validation_data, test_data = cPickle.load(f)
91         training_inputs = get_data(training_data)
92         training_label=[get_label(y_) for y_ in training_data[1]]
93         data = zip(training_inputs,training_label)
94         test_inputs = training_inputs = get_data(test_data)
95         test = zip(test_inputs,test_data[1])
96         net = Network([784, 30, 10])
97         net.SGD(data[:5000],test[:5000],50,10, 3.0,)   # 4476/5000 (4347/5000)