Stanford机器学习笔记-5.神经网络Neural Networks (part two)

5 Neural Networks (part two)

content:

　　5 Neural Networks (part two)

　　　　5.1 cost function

　　　　5.2 Back Propagation

　　　　5.3 神经网络总结

接上一篇4. Neural Networks (part one). 本文将先定义神经网络的代价函数，然后介绍逆向传播(Back Propagation: BP)算法，它能有效求解代价函数对连接权重的偏导，最后对训练神经网络的过程进行总结。

5.1 cost function

(注：正则化相关内容参见3.Bayesian statistics and Regularization)

5.2 Back Propagation

（详细推导过程参见反向传播算法，以及李宏毅的机器学习课程：youtube,B站）。

图5-1 BP算法步骤

在实现反向传播算法时，有如下几个需要注意的地方。

1. 需要对所有的连接权重(包括偏移单元)初始化为接近0但不全等于0的随机数。如果所有参数都用相同的值作为初始值，那么所有隐藏层单元最终会得到与输入值有关的、相同的函数（也就是说，所有神经元的激活值都会取相同的值，对于任何输入x 都会有：  ）。随机初始化的目的是使对称失效。具体地，我们可以如图5-2一样随机初始化。（matlab实现见后文代码1）
2. 如果实现的BP算法计算出的梯度（偏导数）是错误的，那么用该模型来预测新的值肯定是不科学的。所以，我们应该在应用之前就判断BP算法是否正确。具体的，可以通过数值的方法(如图5-3所示的)计算出较精确的偏导，然后再和BP算法计算出来的进行比较，若两者相差在正常的误差范围内，则BP算法计算出的应该是比较正确的，否则说明算法实现有误。注意在检查完后，在真正训练模型时不应该再运行数值计算偏导的方法，否则将会运行很慢。（matlab实现见后文代码2）
3. 用matlab实现时要注意matlab的函数参数不能为矩阵，而连接权重为矩阵，所以在传递初始化连接权重前先将其向量化，再用reshape函数恢复。(见后文代码3)

图5-2 随机初始化连接权重

图5-3 数值方法求代价函数偏导的近似值

5.3 神经网络总结

第一步，设计神经网络结构。

隐藏层单元个数通常都是不确定的。

一般选取神经网络隐藏层单元个数的几个经验公式如下：

参考https://www.zhihu.com/question/46530834

此外，MNIST手写数字识别中给出了以不同的神经网络结构训练的结果，供参考

第二步，实现正向传播(FP)和反向传播算法，这一步包括如下的子步骤。

第三步，用数值方法检查求偏导的正确性

第四步，用梯度下降法或更先进的优化算法求使得代价函数最小的连接权重

在第四步中，由于代价函数是非凸(non-convex)函数，所以在优化过程中可能陷入局部最优值，但不一定比全局最优差很多（如图5-4），在实际应用中通常不是大问题。也会有一些启发式的算法（如模拟退火算法，遗传算法等）来帮助跳出局部最优。

图5-4 陷入局部最优(不一定比全局最优差很多)

 

代码1：随机初始化连接权重

function W = randInitializeWeights(L_in, L_out)
%RANDINITIALIZEWEIGHTS Randomly initialize the weights of a layer with L_in
%incoming connections and L_out outgoing connections
%   W = RANDINITIALIZEWEIGHTS(L_in, L_out) randomly initializes the weights 
%   of a layer with L_in incoming connections and L_out outgoing 
%   connections. 
%
%   Note that W should be set to a matrix of size(L_out, 1 + L_in) as
%   the column row of W handles the "bias" terms
%

W = zeros(L_out, 1 + L_in);


% Instructions: Initialize W randomly so that we break the symmetry while
%               training the neural network.
%
% Note: The first row of W corresponds to the parameters for the bias units
%

epsilon_init = sqrt(6) / (sqrt(L_out+L_in));
W = rand(L_out, 1 + L_in) * 2 * epsilon_init - epsilon_init;

end

代码2：用数值方法求代价函数对连接权重偏导的近似值

function numgrad = computeNumericalGradient(J, theta)
%COMPUTENUMERICALGRADIENT Computes the gradient using "finite differences"
%and gives us a numerical estimate of the gradient.
%   numgrad = COMPUTENUMERICALGRADIENT(J, theta) computes the numerical
%   gradient of the function J around theta. Calling y = J(theta) should
%   return the function value at theta.

% Notes: The following code implements numerical gradient checking, and 
%        returns the numerical gradient.It sets numgrad(i) to (a numerical 
%        approximation of) the partial derivative of J with respect to the 
%        i-th input argument, evaluated at theta. (i.e., numgrad(i) should 
%        be the (approximately) the partial derivative of J with respect 
%        to theta(i).)
%                

numgrad = zeros(size(theta));
perturb = zeros(size(theta));
e = 1e-4;
for p = 1:numel(theta)
    % Set perturbation vector
    perturb(p) = e;
    % Compute Numerical Gradient
    numgrad(p) = ( J(theta + perturb) - J(theta - perturb)) / (2*e);
    perturb(p) = 0;
end
end

代码3：应用FP和BP算法实现计算隐藏层为1层的神经网络的代价函数以及其对连接权重的偏导数

function [J grad] = nnCostFunction(nn_params, ...
                                   input_layer_size, ...
                                   hidden_layer_size, ...
                                   num_labels, ...
                                   X, y, lambda)
%NNCOSTFUNCTION Implements the neural network cost function for a two layer
%neural network which performs classification
%   [J grad] = NNCOSTFUNCTON(nn_params, hidden_layer_size, num_labels, ...
%   X, y, lambda) computes the cost and gradient of the neural network. The
%   parameters for the neural network are "unrolled" into the vector
%   nn_params and need to be converted back into the weight matrices. 
% 
%   The returned parameter grad should be a "unrolled" vector of the
%   partial derivatives of the neural network.
%

% Reshape nn_params back into the parameters Theta1 and Theta2, the weight matrices
% for our 2 layer neural network:Theta1: 1->2; Theta2: 2->3 
Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ...
                 hidden_layer_size, (input_layer_size + 1));
           
Theta2 = reshape(nn_params((1 + (hidden_layer_size * (input_layer_size + 1))):end), ...
                 num_labels, (hidden_layer_size + 1));

% Setup some useful variables
m = size(X, 1);
J = 0;
Theta1_grad = zeros(size(Theta1));  
Theta2_grad = zeros(size(Theta2));

%         Note: The vector y passed into the function is a vector of labels
%               containing values from 1..K. You need to map this vector into a 
%               binary vector of 1's and 0's to be used with the neural network
%               cost function.

for i = 1:m
    % compute activation by Forward Propagation
    a1 = [1; X(i,:)'];
    z2 = Theta1 * a1;
    a2 = [1; sigmoid(z2)];
    z3 = Theta2 * a2;
    h = sigmoid(z3);
    
    yy = zeros(num_labels,1);
    yy(y(i)) = 1;              % 训练集的真实值yy
   
    J = J + sum(-yy .* log(h) - (1-yy) .* log(1-h));
    
    % Back Propagation 
    delta3 = h - yy;
    delta2 = (Theta2(:,2:end)' * delta3) .* sigmoidGradient(z2); %注意要除去偏移单元的连接权重
    
    Theta2_grad = Theta2_grad + delta3 * a2';   
    Theta1_grad = Theta1_grad + delta2 * a1';
end

J = J / m + lambda * (sum(sum(Theta1(:,2:end) .^ 2)) + sum(sum(Theta2(:,2:end) .^ 2))) / (2*m);

Theta2_grad = Theta2_grad / m;
Theta2_grad(:,2:end) = Theta2_grad(:,2:end) + lambda * Theta2(:,2:end) / m; % regularized nn

Theta1_grad = Theta1_grad / m;
Theta1_grad(:,2:end) = Theta1_grad(:,2:end) + lambda * Theta1(:,2:end) / m; % regularized nn

% Unroll gradients
grad = [Theta1_grad(:) ; Theta2_grad(:)];

end