《机器学习》学习笔记(二):神经网络
在解决一些简单的分类问题时,线性回归与逻辑回归就足以应付,但面对更加复杂的问题时(例如对图片中车的类型进行识别),运用之前的线性模型可能就得不到理想的结果,而且由于更大的数据量,之前方法的计算量也会变得异常庞大。因此我们需要学习一个非线性系统:神经网络。
我在学习时,主要通过Andrew Ng教授提供的网络,而且文中多处都有借鉴Andrew Ng教授在mooc提供的资料。
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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。
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posted on 2015-03-05 10:48 chenyn2014 阅读(2420) 评论(0) 编辑 收藏 举报