本周作业：Neural Networks Learning
实现神经网络BP算法，应用于手写数字的辨别。
[*] sigmoidGradient.m - Compute the gradient of the sigmoid function
[*] randInitializeWeights.m - Randomly initialize weights
[*] nnCostFunction.m - Neural network cost function
1.1Feedforward and cost function
CostFunction J（θ）
hθ(x(i))通过Figure2计算得到
nnCostFunction.m
a1 = [ones(m,1) X]; z2 = a1*Theta1'; a2 = sigmoid(z2); a2 = [ones(m,1) a2]; z3 = a2*Theta2'; a3 = sigmoid(z3); h = a3;因为训练的数据集中的y是一个5000*1的十进制数据集，进行数据处理的时候要将它映射为二进制的矩阵yk，一个十进制数用一行01数列表示，其中仅有第n列为1，则该行表示十进制数字n。因此yk为5000*10的一个矩阵。
nnCostFunction.m
yk = zeros(m,num_labels); for i = 1:m yk(i,y(i)) = 1; end计算CostFunction J（θ）且regularized
nnCostFunction.m
J = -1/m*sum(sum(yk.*log(h)+(1-yk).*log(1-h))) + lambda/(2*m)*(sum(sum(Theta1(:,2:end).^2))+sum(sum(Theta2(:,2:end).^2)));计算gradient，用fmincg去minimize J（θ），从而训练神经网络
2.1 Sigmoid gradient
SigmoidGradient.m
g = sigmoid(z).*(1-sigmoid(z));2.2 Random initialization
randInitializeWeights.m
epsilon_init = 0.12; W = rand(L_out,L_in+1)*(2*epsilon_init)-epsilon_init;2.3 Backpropagation
一次计算一个训练集，先用FP算法计算相应的a2,z2,z3,a3,h，再用BP算法计算相应的δ。用for i = 1：m循环对每个训练集进行5个步骤计算。
对第i个训练集计算a2,z2,z3,a3，保证a1,a2要加上一个bias（1）对于第i个训练集的第三层的输出单元k，令对于第二层
累积gradient，对于第二层的δ0要去除，其中l表示第l层的gradient（第一层没有）
计算神经网络的J（θ）的gradient（未正规化）
2.4 Regularized Neural Network
不对θ（l）用于偏移量的第一个元素计算的进行正规化处理
nnCostFunction.m
for i = 1:m a1 = [1,X(i,:)]; z2 = a1*Theta1'; a2 = sigmoid(z2); a2 = [1,a2]; z3 = a2*Theta2'; a3 = sigmoid(z3); h = a3; delta3 = a3 - yk(i,:); delta2 = delta3*Theta2(:,2:end).*sigmoidGradient(z2); grad1 = grad1 + delta2'*a1; grad2 = grad2 + delta3'*a2; end grad1 = 1/m*grad1; grad1(:,2:end) = grad1(:,2:end) + lambda/m*Theta1(:,2:end); grad2 = 1/m*grad2; grad2(:,2:end) = grad2(:,2:end) + lambda/m*Theta2(:,2:end);nnCostFunction.m（全）
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; grad1 = zeros(size(Theta1)); grad2 = 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. % a1 = [ones(m,1) X]; z2 = a1*Theta1'; a2 = sigmoid(z2); a2 = [ones(m,1) a2]; z3 = a2*Theta2'; a3 = sigmoid(z3); h = a3; yk = zeros(m,num_labels); for i = 1:m yk(i,y(i)) = 1; end J = -1/m*sum(sum(yk.*log(h)+(1-yk).*log(1-h))) + lambda/(2*m)*(sum(sum(Theta1(:,2:end).^2))+sum(sum(Theta2(:,2:end).^2))); for i = 1:m a1 = [1,X(i,:)]; z2 = a1*Theta1'; a2 = sigmoid(z2); a2 = [1,a2]; z3 = a2*Theta2'; a3 = sigmoid(z3); h = a3; delta3 = a3 - yk(i,:); delta2 = delta3*Theta2(:,2:end).*sigmoidGradient(z2); grad1 = grad1 + delta2'*a1; grad2 = grad2 + delta3'*a2; end grad1 = 1/m*grad1; grad1(:,2:end) = grad1(:,2:end) + lambda/m*Theta1(:,2:end); grad2 = 1/m*grad2; grad2(:,2:end) = grad2(:,2:end) + lambda/m*Theta2(:,2:end); % ------------------------------------------------------------- % ========================================================================= % Unroll gradients grad = [grad1(:) ; grad2(:)]; end对隐藏层进行可视化处理
displayData(Theta1(:, 2:end));
会发现隐藏层大致对应了在寻找输出层图像的笔画或者其他模式
