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。

 # coding:utf8
 import cPickle
 import numpy as np
 import matplotlib.pyplot as plt

 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:]]  # L(n-1)->L(n)
         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 = self.sigmoid(np.dot(w_, a)+b_)
         return a

     def SGD(self, training_data, test_data,epochs, mini_batch_size, eta):
         n_test = len(test_data)
         n = len(training_data)
         plt.xlabel('epoch')
         plt.title('cost')
         cy=[]
         cx=range(epochs)
         for j in cx:
             self.cost = 0.0
             np.random.shuffle(training_data)  # shuffle
             for k in xrange(0, n, mini_batch_size):
                 mini_batch = training_data[k:k+mini_batch_size]
                 self.update_mini_batch(mini_batch, eta)
             cy.append(self.cost/n)
             print "Epoch {0}: {1} / {2}".format(
                     j, self.evaluate(test_data), n_test)
         plt.plot(cx,cy)
         plt.scatter(cx,cy)
         plt.show()

     def update_mini_batch(self, mini_batch, eta):
         for x, y in mini_batch:
             delta_b, delta_w,cost = self.backprop(x, y)
             self.weights -= eta/len(mini_batch)*delta_w
             self.biases -= eta/len(mini_batch)*delta_b
             self.cost += cost

     def backprop(self, x, y):
         b=np.zeros_like(self.biases)
         w=np.zeros_like(self.weights)
         a_ = x
         a = [x]
         for b_, w_ in zip(self.biases, self.weights):
             a_ = self.sigmoid(np.dot(w_, a_)+b_)
             a.append(a_)
         for l in xrange(1, self.num_layers):
             if l==1:
                 # delta= self.sigmoid_prime(a[-1])*(a[-1]-y)  # O(k)=a[-1], t(k)=y
                 delta= a[-1]-y  # cross-entropy
             else:
                 sp = self.sigmoid_prime(a[-l])   # O(j)=a[-l]
                 delta = np.dot(self.weights[-l+1].T, delta) * sp
             b[-l] = delta
             w[-l] = np.dot(delta, a[-l-1].T)
         cost=0.5*np.sum((b[-1])**2)
         return (b, w,cost)

     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 sigmoid(self,z):
         return 1.0/(1.0+np.exp(-z))

     def sigmoid_prime(self,z):
         return z*(1-z)

 if __name__ == '__main__':

         def get_label(i):
             c=np.zeros((10,1))
             c[i]=1
             return c

         def get_data(data):
             return [np.reshape(x, (784,1)) for x in data[0]]

         f = open('mnist.pkl', 'rb')
         training_data, validation_data, test_data = cPickle.load(f)
         training_inputs = get_data(training_data)
         training_label=[get_label(y_) for y_ in training_data[1]]
         data = zip(training_inputs,training_label)
         test_inputs = training_inputs = get_data(test_data)
         test = zip(test_inputs,test_data[1])
         net = Network([784, 30, 10])
         net.SGD(data[:5000],test[:5000],50,10, 3.0,)   # 4476/5000 (4347/5000)