本次文章內容： Coursera吳恩達機器學習課程，第五週程式設計作業。程式語言是Matlab。

學習演算法分兩部分進行理解，第一部分是根據code對演算法進行綜述，第二部分是程式碼。

0 Introduction 

在這個練習中，將應用 backpropagation,實現神經網路來識別手寫數字。

1 Neural Networks

Algorithm 

Part 0 Initialization & Setup the parameters

         clear ; close all; clc 的三劍客常規操作。並設定神經網路的結構引數。

         下圖是我們的Model representation：一共是三層，an input layer,a hidden layer and an output layer.

Part 1: Loading and Visualizing Data

           隨機選取100個data進行視覺化。

Part 2: Loading Parameters

          載入已經訓練好的theta引數，並將theta展開為我們需要的向量形式。

Part 3: Compute Cost (Feedforward)

         要做back propagation，先要做feedforward得到每個class的hypothesis。這部分的程式碼主要整合在 nnCostFunction（）函式中。 根據我們的model，計算得到每個note的activation值，計算cost function（帶正則項）。next，我們會在這個函式裡計算back propagation，從後向前計算detla value，then計算累積誤差，then計算cost function的partial derivative（帶正則項）。

Part 4: Implement Regularization

         剛剛說到了，並輸入測試值檢測是否得出正確值。

Part 5: Sigmoid Gradient 

          sigmoid 函式的derivative有個特殊性質，這裡計算其導數。

Part 6: Initializing Pameters 

         為了防止所有的theta的初始值都是零，或者same，這將會倒是 back propagation 的計算無意義。所以我們random initialization ，根據維度rand() 0或1 value。

 Part 7: Implement Backpropagation

按4步，計算cost function的gradient，

          1. 計算forward propagation，computing the activations for all layers，注意bias的 +1term。

          2.從output開始從後往前計算detla值

          3.計算hidden layer 的detla 值，注意skip 第一層的detla 值

          4.Accumulate all the gradient值

          5. Obtain the (unregularized) gradient for cost function。

 注意，只有 成功  completed the feedforward之後，才可以implementing the backpropagation algorithm,

Part 8: Implement Regularization

            上一步中，新增正則項，在求cost function的gradient得第五步時。

Part 8: Training NN

         在我們獲得了cost和gradient之後，我們可以通過fmincg（）函式來train neural network。

Part 9: Visualize Weights

         對中間層的畫素，進行視覺化。

Part 10: Implement Predict 

         After training the neural network，我們 計算 training set accuracy。

Code  

下面是nnCostFunction（）函式部分，其他部分難度不大。

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.
%

%me******************************************************
%X= [ones(m,1)  X];
%a1 = sigmoid(X * Theta1');

%m1 = size(a1 ,1);
%X1 = [ones(m1,1) a1];
%hx = sigmoid(X1 * Theta2');

%[p pp] = max(hx, [], 1);
%J1 = (-y).* log(pp) - (1-y).* log(1-pp);
%J = 1/m * sum(J1);

%deta1 = pp -y;
%teacher&***********************************************

ylable = zeros(num_labels, m);   %10x5000
for i = 1:m
    ylable(y(i),i) = 1;
end

a1 = [ones(m,1) X];    %5000x401
z2 = a1 * Theta1';     %5000x25
a2 = sigmoid(z2);     

a2 = [ones(m,1) a2];      %5000x26
a3 = sigmoid(a2 * Theta2');        %5000x10


J = 1 / m * sum( sum( -ylable'.* log(a3) -  (1-ylable').*log(1-a3) )); 

% pay attention :" Theta1(:,2:end) " , no "Theta1" .  
regularized = lambda/(2*m) * (sum(sum(Theta1(:,2:end).^2)) + sum(sum(Theta2(:,2:end).^2)) ); 

J = J + regularized;

delta3 = a3-ylable';            %5000x10 

delta2 = delta3 * Theta2 ;  %5000x26
delta2 = delta2(:,2:end);    %5000x25
delta2 = delta2 .* sigmoidGradient(z2);  %5000x25 

Delta_1 = zeros(size(Theta1));        %25x401
Delta_2 = zeros(size(Theta2));         %10x26

Delta_1 = Delta_1 + delta2' * a1 ;    %25x401
Delta_2 = Delta_2 + delta3' * a2 ;     %10x26

Theta1_grad = 1/m * Delta_1;   
Theta2_grad = 1/m * Delta_2;  

% do the regularization
regularized_1 = lambda/m * Theta1;  
regularized_2 = lambda/m * Theta2;  
% j = 0 is not need to do the regularization
regularized_1(:,1) = zeros(size(regularized_1,1),1);  
regularized_2(:,1) = zeros(size(regularized_2,1),1);  
  
Theta1_grad = Theta1_grad + regularized_1;  
Theta2_grad = Theta2_grad + regularized_2; 

% -------------------------------------------------------------

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

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


end