神经网络算法学习之梯度下降算法

神经网络算法学习之梯度下降算法

1.FeedForward Network: 神经网络中没有循环, 信息单项向前传递

2.识别手写数字，分层—segmentation分层得到minist数据中每一张图片的信息

MNIST dataset: http://yann.lecun.com/exdb/mnist/

3.每个隐藏曾学到不一样的东西

4.用（gradient descent）梯度下降算法Cost函数最小化

C: cost

w: weight 权重

b: bias 偏向

n: 训练数据集实例个数

x: 输入值

a: 输出值 (当x是输入时)

||v||:  向量的length function

C(w,b) 越小越好，输出的预测值和真实值差别越小

5.更新wk和bl

6.更具体直接私信我

相关代码

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.weight = [np.random.randn(y,x) for x,y in zip(sizes[:-1],sizes[1:])]

    def feedforward(self,a):
        """
        return the output of the network if "a" is input.
        :param a: input vector
        :return: output vector
        """
        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):
        """
        Train the neural network using mini-batch stochastic gradient descent
        :param training_data: a list of tuples "(x,y)"
        :param epochs: number of testing
        :param mini_batch_size:
        :param eta: learning rate
        :param test_data:
        :return:
        """
        if test_data:
            n_test = len(test_data)
        n = len(training_data)
        for j in xrange(epochs):
            random.shuffle(training_data)
            mini_batchs = [training_data[k:k+mini_batch_size] for k in xrange(0,n,mini_batch_size)]
            for mini_batch in mini_batchs:
                self.update_mini_batch(mini_batch,eta)
            if test_data:
                print "Epoch {0}:{1}/{2}".format(j,self.evaluate(test_data),n_test)
            else:
                print "Epoch {0} complete".format(j)

    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
        :param mini_batch: each batch
        :param eta: learning rate
        :return:
        """
        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):
        """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 evaluate(self, test_data):
        """Return the number of test inputs for which the neural
        network outputs the correct result. Note that the neural
        network's output is assumed to be the index of whichever
        neuron in the final layer has the highest activation."""
        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 the vector of partial derivatives \partial C_x /
        \partial a for the output activations."""
        return (output_activations-y)

#### Miscellaneous functions
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))