转 神经网络

神经网络在解决一些复杂的非线性分类问题时，相对于线性回归、逻辑回归，都被证明是一个更好的算法。其实神经网络也可以看做的逻辑回归的组合（叠加，级联等）。

        一个典型神经网络的模型如下图所示：

                 

        上述模型由3个部分组成：输入层、隐藏层、输出层。其中输入层输入特征值，输出层的输出作为我们分类的依据。例如一个20*20大小的手写数字图片的识别举例，那么输入层的输入便可以是20*20=400个像素点的像素值，即模型中的a1；输出层的输出便可以看做是该幅图片是0到9其中某个数字的概率。而隐藏层、输出层中的每个节点其实都可以看做是逻辑回归得到的。逻辑回归的模型可以看做这样（如下图所示）：

                        

        有了神经网络的模型，我们的目的就是求解模型里边的参数theta，为此我们还需知道该模型的代价函数以及每一个节点的“梯度值”。

        代价函数的定义如下：

                                               

      代价函数关于每一个节点处theta的梯度可以用反向传播算法计算出来。反向传播算法的思想是由于我们无法直观的得到隐藏层的输出，但我们已知输出层的输出，通过反向传播，倒退其参数。

我们以以下模型举例，来说明反向传播的思路、过程：

                      

该模型与给出的第一个模型不同的是，它具有两个隐藏层。

        为了熟悉这个模型，我们需要先了解前向传播的过程，对于此模型，前向传播的过程如下：

                  

其中，a1，z2等参数的意义可以参照本文给出的第一个神经网络模型，类比得出。

然后我们定义误差delta符号具有如下含义（之后推导梯度要用）：

               

误差delta的计算过程如下：

             

然后我们通过反向传播算法求得节点的梯度，反向传播算法的过程如下：

      

有了代价函数与梯度函数，我们可以先用数值的方法检测我们的梯度结果。之后我们就可以像之前那样调用matlab的fminunc函数求得最优的theta参数。

需要注意的是，在初始化theta参数时，需要赋予theta随机值，而不能是固定为0或是什么，这就避免了训练之后，每个节点的参数都是一样的。

下面给出计算代价与梯度的代码：

 

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 = 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);

% You need to return the following variables correctly
J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));

% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the code by working through the
%               following parts.
%
% Part 1: Feedforward the neural network and return the cost in the
%         variable J. After implementing Part 1, you can verify that your
%         cost function computation is correct by verifying the cost
%         computed in ex4.m
%
% Part 2: Implement the backpropagation algorithm to compute the gradients
%         Theta1_grad and Theta2_grad. You should return the partial derivatives of
%         the cost function with respect to Theta1 and Theta2 in Theta1_grad and
%         Theta2_grad, respectively. After implementing Part 2, you can check
%         that your implementation is correct by running checkNNGradients
%
%         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.
%
%         Hint: We recommend implementing backpropagation using a for-loop
%               over the training examples if you are implementing it for the
%               first time.
%
% Part 3: Implement regularization with the cost function and gradients.
%
%         Hint: You can implement this around the code for
%               backpropagation. That is, you can compute the gradients for
%               the regularization separately and then add them to Theta1_grad
%               and Theta2_grad from Part 2.
%
J_tmp=zeros(m,1);
for i=1:m
    y_vec=zeros(num_labels,1);
    y_vec(y(i))=1;
    a1 = [ones(1, 1) X(i,:)]';
    z2=Theta1*a1;
    a2=sigmoid(z2);
    a2=[ones(1,size(a2,2)); a2];
    z3=Theta2*a2;
    a3=sigmoid(z3);
    hThetaX=a3;
    J_tmp(i)=sum(-y_vec.*log(hThetaX)-(1-y_vec).*log(1-hThetaX));
end
J=1/m*sum(J_tmp);
J=J+lambda/(2*m)*(sum(sum(Theta1(:,2:end).^2))+sum(sum(Theta2(:,2:end).^2)));

Delta1 = zeros( hidden_layer_size, (input_layer_size + 1));
Delta2 = zeros( num_labels, (hidden_layer_size + 1));
for t=1:m
    y_vec=zeros(num_labels,1);
    y_vec(y(t))=1;
    a1 = [1 X(t,:)]';
    z2=Theta1*a1;
    a2=sigmoid(z2);
    a2=[ones(1,size(a2,2)); a2];
    z3=Theta2*a2;
    a3=sigmoid(z3);
    delta_3=a3-y_vec;
    gz2=[0;sigmoidGradient(z2)];
    delta_2=Theta2'*delta_3.*gz2;
    delta_2=delta_2(2:end);
    Delta2=Delta2+delta_3*a2';
    Delta1=Delta1+delta_2*a1';
end
Theta1_grad=1/m*Delta1;
Theta2_grad=1/m*Delta2;

Theta1(:,1)=0;
Theta1_grad=Theta1_grad+lambda/m*Theta1;
Theta2(:,1)=0;
Theta2_grad=Theta2_grad+lambda/m*Theta2;
% -------------------------------------------------------------

% =========================================================================

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


end

最后总结一下，对于一个典型的神经网络，训练过程如下：

 

     

   

按照这个步骤，我们就可以求得神经网络的参数theta。

转载请注明出处：http://blog.csdn.net/u010278305