本週作業：Neural Networks Learning
實現神經網絡BP算法，應用於手寫數字的辨別。
[*] 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))通過Figure2計算得到
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;因爲訓練的數據集中的y是一個5000*1的十進制數據集，進行數據處理的時候要將它映射爲二進制的矩陣yk，一個十進制數用一行01數列表示，其中僅有第n列爲1，則該行表示十進制數字n。因此yk爲5000*10的一個矩陣。
nnCostFunction.m
yk = zeros(m,num_labels);
for i = 1:m
yk(i,y(i)) = 1;
end
計算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
計算gradient，用fmincg去minimize J（θ），從而訓練神經網絡
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
一次計算一個訓練集，先用FP算法計算相應的a2,z2,z3,a3,h，再用BP算法計算相應的δ。用for i = 1：m循環對每個訓練集進行5個步驟計算。
- 對第i個訓練集計算a2,z2,z3,a3，保證a1,a2要加上一個bias（1）
- 對於第i個訓練集的第三層的輸出單元k，令
- 對於第二層
- 累積gradient，對於第二層的δ0要去除，其中l表示第l層的gradient（第一層沒有）
- 計算神經網絡的J（θ）的gradient（未正規化）
2.4 Regularized Neural Network
不對θ（l）用於偏移量的第一個元素計算的進行正規化處理
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
對隱藏層進行可視化處理
displayData(Theta1(:, 2:end));
會發現隱藏層大致對應了在尋找輸出層圖像的筆畫或者其他模式