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.Neural Network
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)));
2.Backpropagation
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
3.Visualizing the hidden layer
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