【Exercise 4 Neural Network Learning】
【代碼】
ex4.m
-> 設置網絡結構
-> 隨機展示100個樣本
sel = randperm(size(X, 1));
sel = sel(1:100);
displayData(X(sel, :));
-> 實現成本函數及其正則化（前向傳播）
實現sigmoid的導數
-> 隨機初始化θ 並變成列向量
-> 實現梯度函數及其正則化（反向傳播）
-> 數值法驗證梯度函數實現
正則化再驗證
-> 調包最優化 θ恢復爲矩陣
-> 第一層權重可視化 （僅 2：end 的權重與m個像素/輸入相連，截距項沒有）
-> 預測、計算準確率
%% Machine Learning Online Class - Exercise 4 Neural Network Learning
% Instructions
% ------------
%
% This file contains code that helps you get started on the
% linear exercise. You will need to complete the following functions
% in this exericse:
%
% sigmoidGradient.m
% randInitializeWeights.m
% nnCostFunction.m
%
% For this exercise, you will not need to change any code in this file,
% or any other files other than those mentioned above.
%
%% Initialization
clear ; close all; clc
%% Setup the parameters you will use for this exercise
input_layer_size = 400; % 20x20 Input Images of Digits
hidden_layer_size = 25; % 25 hidden units
num_labels = 10; % 10 labels, from 1 to 10
% (note that we have mapped "0" to label 10)
%% =========== Part 1: Loading and Visualizing Data =============
% We start the exercise by first loading and visualizing the dataset.
% You will be working with a dataset that contains handwritten digits.
%
% Load Training Data
fprintf('Loading and Visualizing Data ...\n')
load('ex4data1.mat');
m = size(X, 1);
% Randomly select 100 data points to display
sel = randperm(size(X, 1));
sel = sel(1:100);
displayData(X(sel, :));
fprintf('Program paused. Press enter to continue.\n');
pause;
%% ================ Part 2: Loading Parameters ================
% In this part of the exercise, we load some pre-initialized
% neural network parameters.
fprintf('\nLoading Saved Neural Network Parameters ...\n')
% Load the weights into variables Theta1 and Theta2
load('ex4weights.mat');
% Unroll parameters
nn_params = [Theta1(:) ; Theta2(:)];
%% ================ Part 3: Compute Cost (Feedforward) ================
% To the neural network, you should first start by implementing the
% feedforward part of the neural network that returns the cost only. You
% should complete the code in nnCostFunction.m to return cost. After
% implementing the feedforward to compute the cost, you can verify that
% your implementation is correct by verifying that you get the same cost
% as us for the fixed debugging parameters.
%
% We suggest implementing the feedforward cost *without* regularization
% first so that it will be easier for you to debug. Later, in part 4, you
% will get to implement the regularized cost.
%
fprintf('\nFeedforward Using Neural Network ...\n')
% Weight regularization parameter (we set this to 0 here).
lambda = 0;
J = nnCostFunction(nn_params, input_layer_size, hidden_layer_size, ...
num_labels, X, y, lambda);
fprintf(['Cost at parameters (loaded from ex4weights): %f '...
'\n(this value should be about 0.287629)\n'], J);
fprintf('\nProgram paused. Press enter to continue.\n');
pause;
%% =============== Part 4: Implement Regularization ===============
% Once your cost function implementation is correct, you should now
% continue to implement the regularization with the cost.
%
fprintf('\nChecking Cost Function (w/ Regularization) ... \n')
% Weight regularization parameter (we set this to 1 here).
lambda = 1;
J = nnCostFunction(nn_params, input_layer_size, hidden_layer_size, ...
num_labels, X, y, lambda);
fprintf(['Cost at parameters (loaded from ex4weights): %f '...
'\n(this value should be about 0.383770)\n'], J);
fprintf('Program paused. Press enter to continue.\n');
pause;
%% ================ Part 5: Sigmoid Gradient ================
% Before you start implementing the neural network, you will first
% implement the gradient for the sigmoid function. You should complete the
% code in the sigmoidGradient.m file.
%
fprintf('\nEvaluating sigmoid gradient...\n')
g = sigmoidGradient([-1 -0.5 0 0.5 1]);
fprintf('Sigmoid gradient evaluated at [-1 -0.5 0 0.5 1]:\n ');
fprintf('%f ', g);
fprintf('\n\n');
fprintf('Program paused. Press enter to continue.\n');
pause;
%% ================ Part 6: Initializing Pameters ================
% In this part of the exercise, you will be starting to implment a two
% layer neural network that classifies digits. You will start by
% implementing a function to initialize the weights of the neural network
% (randInitializeWeights.m)
fprintf('\nInitializing Neural Network Parameters ...\n')
initial_Theta1 = randInitializeWeights(input_layer_size, hidden_layer_size);
initial_Theta2 = randInitializeWeights(hidden_layer_size, num_labels);
% Unroll parameters
initial_nn_params = [initial_Theta1(:) ; initial_Theta2(:)];
%% =============== Part 7: Implement Backpropagation ===============
% Once your cost matches up with ours, you should proceed to implement the
% backpropagation algorithm for the neural network. You should add to the
% code you've written in nnCostFunction.m to return the partial
% derivatives of the parameters.
%
fprintf('\nChecking Backpropagation... \n');
% Check gradients by running checkNNGradients
checkNNGradients;
fprintf('\nProgram paused. Press enter to continue.\n');
pause;
%% =============== Part 8: Implement Regularization ===============
% Once your backpropagation implementation is correct, you should now
% continue to implement the regularization with the cost and gradient.
%
fprintf('\nChecking Backpropagation (w/ Regularization) ... \n')
% Check gradients by running checkNNGradients
lambda = 3;
checkNNGradients(lambda);
% Also output the costFunction debugging values
debug_J = nnCostFunction(nn_params, input_layer_size, ...
hidden_layer_size, num_labels, X, y, lambda);
fprintf(['\n\nCost at (fixed) debugging parameters (w/ lambda = %f): %f ' ...
'\n(for lambda = 3, this value should be about 0.576051)\n\n'], lambda, debug_J);
fprintf('Program paused. Press enter to continue.\n');
pause;
%% =================== Part 8: Training NN ===================
% You have now implemented all the code necessary to train a neural
% network. To train your neural network, we will now use "fmincg", which
% is a function which works similarly to "fminunc". Recall that these
% advanced optimizers are able to train our cost functions efficiently as
% long as we provide them with the gradient computations.
%
fprintf('\nTraining Neural Network... \n')
% After you have completed the assignment, change the MaxIter to a larger
% value to see how more training helps.
options = optimset('MaxIter', 50);
% You should also try different values of lambda
lambda = 1;
% Create "short hand" for the cost function to be minimized
costFunction = @(p) nnCostFunction(p, ...
input_layer_size, ...
hidden_layer_size, ...
num_labels, X, y, lambda);
% Now, costFunction is a function that takes in only one argument (the
% neural network parameters)
[nn_params, cost] = fmincg(costFunction, initial_nn_params, options);
% Obtain Theta1 and Theta2 back from nn_params
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));
fprintf('Program paused. Press enter to continue.\n');
pause;
%% ================= Part 9: Visualize Weights =================
% You can now "visualize" what the neural network is learning by
% displaying the hidden units to see what features they are capturing in
% the data.
fprintf('\nVisualizing Neural Network... \n')
displayData(Theta1(:, 2:end));
fprintf('\nProgram paused. Press enter to continue.\n');
pause;
%% ================= Part 10: Implement Predict =================
% After training the neural network, we would like to use it to predict
% the labels. You will now implement the "predict" function to use the
% neural network to predict the labels of the training set. This lets
% you compute the training set accuracy.
pred = predict(Theta1, Theta2, X);
fprintf('\nTraining Set Accuracy: %f\n', mean(double(pred == y)) * 100);
displayData.m
一張圖片——X的一行
m：圖片數
n：一張圖片的像素數
-> 圖片橫向尺寸(example_width)可以作爲參數傳入，未傳入則認爲是正方形，像素總數開平方求得；再求得縱向尺寸(example_height)
-> 每行展示圖片張數(display_rows)由圖片總數開平方求得；再求得縱向張數(display_cols)
-> 根據以上數據生成空“畫板”display_array
-> 逐行從X取出圖片、reshape、粘貼至display_array
-> imagesc函數展示
function [h, display_array] = displayData(X, example_width)
%DISPLAYDATA Display 2D data in a nice grid
% [h, display_array] = DISPLAYDATA(X, example_width) displays 2D data
% stored in X in a nice grid. It returns the figure handle h and the
% displayed array if requested.
% Set example_width automatically if not passed in
if ~exist('example_width', 'var') || isempty(example_width)
example_width = round(sqrt(size(X, 2)));
end
% Gray Image
colormap(gray);
% Compute rows, cols
[m n] = size(X);
example_height = (n / example_width);
% Compute number of items to display
display_rows = floor(sqrt(m));
display_cols = ceil(m / display_rows);
% Between images padding
pad = 1;
% Setup blank display
display_array =255 - ones(pad + display_rows * (example_height + pad), ...
pad + display_cols * (example_width + pad));
% Copy each example into a patch on the display array
curr_ex = 1;
for j = 1:display_rows
for i = 1:display_cols
if curr_ex > m,
break;
end
% Copy the patch
% Get the max value of the patch
max_val = max(abs(X(curr_ex, :)));
display_array(pad + (j - 1) * (example_height + pad) + (1:example_height), ...
pad + (i - 1) * (example_width + pad) + (1:example_width)) = ...
reshape(X(curr_ex, :), example_height, example_width) / max_val;
curr_ex = curr_ex + 1;
end
if curr_ex > m,
break;
end
end
% Display Image
h = imagesc(display_array, [-1 1]);
% Do not show axis
axis image off
drawnow;
end
nnCostFuction.m
核心部分：前向傳播預測、算成本；反向傳播算梯度。
1、y重新編碼
2、前向傳播
X轉置爲列向量的堆疊（約定俗成且更方便的形式）
接下來可以套公式了
note： - 加截距項a_0時要加1×m個1
- sigmoid函數最開始的實現就考慮了通用性：對向量和矩陣要element(entry)-wise運算，故公式形式不變
3、成本
非正則項即對m個樣本、K個分類器的損失函數取平均，而兩層求和正好對應X、A、Z等矩陣的尺寸：label_num×m
那麼只要對這些矩陣進行某些點運算（element-wise），得到某個損失矩陣L，它的元素L(i,k)恰好爲第i個元素第k個分類器的損失函數。則只需對矩陣L求和。
正則項套公式即可，注意截距項特例。
4、反向傳播
（1）δ反向傳播
（2）利用δ計算梯度
接下來可以套公式了，同樣正則化注意截距項特例
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.
%
%recode y from (0,1,2,3...) to ([0001],[0010],...)
Y=zeros(num_labels,m);
for i =1:length(y)
Y(y(i),i)=1;
end
%transpose to the more common way
X=X';
%forwarding
A1 = [ones(1,m);X];
Z2 = Theta1*A1;
A2 = sigmoid(Z2);
A2 = [ones(1,m);A2];
Z3 = Theta2*A2;
A3 = sigmoid(Z3);
%compute J
J = -1/m*sum(sum( Y.*log(A3) + (1-Y).*log(1-A3) ));
J = J + lambda/2/m*( sum(sum((Theta1(:,2:end)).^2)) + sum(sum((Theta2(:,2:end)).^2)) );
%backwarding
lowercase_delta_3 = A3-Y;
lowercase_delta_2 = Theta2(:,2:end)'*lowercase_delta_3.*sigmoidGradient(Z2);
capital_delta_1 = lowercase_delta_2*A1';
capital_delta_2 = lowercase_delta_3*A2';
Theta1_grad = capital_delta_1/m;
Theta2_grad = capital_delta_2/m;
Theta1_grad(:,2:end)=Theta1_grad(:,2:end)+lambda/m*Theta1(:,2:end);
Theta2_grad(:,2:end)=Theta2_grad(:,2:end)+lambda/m*Theta2(:,2:end);
% -------------------------------------------------------------
% =========================================================================
% Unroll gradients
grad = [Theta1_grad(:) ; Theta2_grad(:)];
end整體：復原爲矩陣（方便向量化計算） -> 核心部分 -> unroll爲列向量（可以送入優化函數）
sigmoidGradient.m
套公式，注意element-wise
function g = sigmoidGradient(z)
%SIGMOIDGRADIENT returns the gradient of the sigmoid function
%evaluated at z
% g = SIGMOIDGRADIENT(z) computes the gradient of the sigmoid function
% evaluated at z. This should work regardless if z is a matrix or a
% vector. In particular, if z is a vector or matrix, you should return
% the gradient for each element.
g = zeros(size(z));
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the gradient of the sigmoid function evaluated at
% each value of z (z can be a matrix, vector or scalar).
g=sigmoid(z).*(1-sigmoid(z));
% =============================================================
end
randInitializeWeights.m
隨機小值初始化
通過epsilon設置小值範圍：[-epsilon , epsilon ]
Theta大小爲
function W = randInitializeWeights(L_in, L_out)
%RANDINITIALIZEWEIGHTS Randomly initialize the weights of a layer with L_in
%incoming connections and L_out outgoing connections
% W = RANDINITIALIZEWEIGHTS(L_in, L_out) randomly initializes the weights
% of a layer with L_in incoming connections and L_out outgoing
% connections.
%
% Note that W should be set to a matrix of size(L_out, 1 + L_in) as
% the first column of W handles the "bias" terms
%
% You need to return the following variables correctly
W = zeros(L_out, 1 + L_in);
% ====================== YOUR CODE HERE ======================
% Instructions: Initialize W randomly so that we break the symmetry while
% training the neural network.
%
% Note: The first column of W corresponds to the parameters for the bias unit
%
% Randomly initialize the weights to small values
epsilon_init = 0.12;
W = rand(L_out, 1 + L_in) * 2 * epsilon_init - epsilon_init;
% =========================================================================
end
checkNNGradients.m
數值法驗算梯度
知道J就可以估算——不論正則化與否，同樣方法驗算
此外nnCostFunction傳遞了網絡結構參數，所以可以適應不同網絡，可以用一個簡單網絡驗算（但網絡層數固定）
-> 構建簡單網絡
-> 僞隨機初始化θ 並變成列向量（固定計算式，固定初始值，方便調試）
-> 生成X,y
-> 反向傳播計算、數值計算
-> 同時列出比較，計算誤差（公式：norm(差)/norm(和) 有什麼特別含義嗎？）
function checkNNGradients(lambda)
%CHECKNNGRADIENTS Creates a small neural network to check the
%backpropagation gradients
% CHECKNNGRADIENTS(lambda) Creates a small neural network to check the
% backpropagation gradients, it will output the analytical gradients
% produced by your backprop code and the numerical gradients (computed
% using computeNumericalGradient). These two gradient computations should
% result in very similar values.
%
if ~exist('lambda', 'var') || isempty(lambda)
lambda = 0;
end
input_layer_size = 3;
hidden_layer_size = 5;
num_labels = 3;
m = 5;
% We generate some 'random' test data
Theta1 = debugInitializeWeights(hidden_layer_size, input_layer_size);
Theta2 = debugInitializeWeights(num_labels, hidden_layer_size);
% Reusing debugInitializeWeights to generate X
X = debugInitializeWeights(m, input_layer_size - 1);
y = 1 + mod(1:m, num_labels)';
% Unroll parameters
nn_params = [Theta1(:) ; Theta2(:)];
% Short hand for cost function
costFunc = @(p) nnCostFunction(p, input_layer_size, hidden_layer_size, ...
num_labels, X, y, lambda);
[cost, grad] = costFunc(nn_params);
numgrad = computeNumericalGradient(costFunc, nn_params);
% Visually examine the two gradient computations. The two columns
% you get should be very similar.
disp([numgrad grad]);
fprintf(['The above two columns you get should be very similar.\n' ...
'(Left-Your Numerical Gradient, Right-Analytical Gradient)\n\n']);
% Evaluate the norm of the difference between two solutions.
% If you have a correct implementation, and assuming you used EPSILON = 0.0001
% in computeNumericalGradient.m, then diff below should be less than 1e-9
diff = norm(numgrad-grad)/norm(numgrad+grad);
fprintf(['If your backpropagation implementation is correct, then \n' ...
'the relative difference will be small (less than 1e-9). \n' ...
'\nRelative Difference: %g\n'], diff);
enddebugInitializeWeights.m
按固定表達式生成僞隨機初始權重，固定值，方便調試
numel——元素個數
function W = debugInitializeWeights(fan_out, fan_in)
%DEBUGINITIALIZEWEIGHTS Initialize the weights of a layer with fan_in
%incoming connections and fan_out outgoing connections using a fixed
%strategy, this will help you later in debugging
% W = DEBUGINITIALIZEWEIGHTS(fan_in, fan_out) initializes the weights
% of a layer with fan_in incoming connections and fan_out outgoing
% connections using a fix set of values
%
% Note that W should be set to a matrix of size(1 + fan_in, fan_out) as
% the first row of W handles the "bias" terms
%
% Set W to zeros
W = zeros(fan_out, 1 + fan_in);
% Initialize W using "sin", this ensures that W is always of the same
% values and will be useful for debugging
W = reshape(sin(1:numel(W)), size(W)) / 10;
% =========================================================================
end
computeNumericalGradient.m
數值計算梯度， 對向量求近似偏導數：每個分量分別計算
perturb參數擾動
e擾動大小
對每個分量：
{
形成擾動
計算參數分別加減擾動時成本
代公式
擾動恢復爲全零，爲下個分量做準備
}
note：函數作爲參數傳遞，可以參考
function numgrad = computeNumericalGradient(J, theta)
%COMPUTENUMERICALGRADIENT Computes the gradient using "finite differences"
%and gives us a numerical estimate of the gradient.
% numgrad = COMPUTENUMERICALGRADIENT(J, theta) computes the numerical
% gradient of the function J around theta. Calling y = J(theta) should
% return the function value at theta.
% Notes: The following code implements numerical gradient checking, and
% returns the numerical gradient.It sets numgrad(i) to (a numerical
% approximation of) the partial derivative of J with respect to the
% i-th input argument, evaluated at theta. (i.e., numgrad(i) should
% be the (approximately) the partial derivative of J with respect
% to theta(i).)
%
numgrad = zeros(size(theta));
perturb = zeros(size(theta));
e = 1e-4;
for p = 1:numel(theta)
% Set perturbation vector
perturb(p) = e;
loss1 = J(theta - perturb);
loss2 = J(theta + perturb);
% Compute Numerical Gradient
numgrad(p) = (loss2 - loss1) / (2*e);
perturb(p) = 0;
end
end
predict.m
與ex3一致：前向傳播，最大值，返回索引
function p = predict(Theta1, Theta2, X)
%PREDICT Predict the label of an input given a trained neural network
% p = PREDICT(Theta1, Theta2, X) outputs the predicted label of X given the
% trained weights of a neural network (Theta1, Theta2)
% Useful values
m = size(X, 1);
num_labels = size(Theta2, 1);
% You need to return the following variables correctly
p = zeros(size(X, 1), 1);
h1 = sigmoid([ones(m, 1) X] * Theta1');
h2 = sigmoid([ones(m, 1) h1] * Theta2');
[dummy, p] = max(h2, [], 2);
% =========================================================================
end