深度学习 Deep Learning UFLDL 最新 Tutorial 学习笔记 1：Linear Regression

1 前言

Andrew Ng的UFLDL在2014年9月底更新了。

对于開始研究Deep Learning的童鞋们来说这真的是极大的好消息!

新的Tutorial相比旧的Tutorial添加了Convolutional Neural Network的内容。了解的童鞋都知道CNN在Computer Vision的重大影响。

而且从新编排了内容及exercises。

新的UFLDL网址为：

http://ufldl.stanford.edu/tutorial/

2 Linear Regression 理论简述

对于线性回归Linear Regression，恐怕大部分童鞋都了解。简单的说

线性回归问题就是一个目标值y取决于一组输入值x。我们要寻找一个最合适的如果Hypothesis来描写叙述这个y与x的关系。然后利用这个Hypothesis来预測新的输入x相应的y。

这是个简单的最优化问题。我们须要一个代价函数cost function来描写叙述在training set样本中的y与通过h函数预測的y之间的差距，从而利用这个cost function通过Gradient Decent梯度下降法来计算h的最优參数从而得到最优的h。

由于是通过样本让计算机“学习”合适的參数theta，因此这是一个最主要的机器学习算法。

cost function：

J(θ)=12∑i(hθ(x(i))−y(i))2=12∑i(θ⊤x(i)−y(i))2

对theta做偏导：

Differentiating the cost function J(θ) as given above with respect to a particular parameter θj gives us:

∂J(θ)∂θj=∑ix(i)j(hθ(x(i))−y(i))

3 Linear Regression 练习

3.1 ex1a_linreg.m 分析

%
%This exercise uses a data from the UCI repository:
% Bache, K. & Lichman, M. (2013). UCI Machine Learning Repository
% http://archive.ics.uci.edu/ml
% Irvine, CA: University of California, School of Information and Computer Science.
%
%Data created by:
% Harrison, D. and Rubinfeld, D.L.
% ''Hedonic prices and the demand for clean air''
% J. Environ. Economics & Management, vol.5, 81-102, 1978.
%
addpath ../common
addpath ../common/minFunc_2012/minFunc
addpath ../common/minFunc_2012/minFunc/compiled

% Load housing data from file.
data = load('housing.data');  % housing data  506x14 
data=data'; % put examples in columns  14x506  一般这里将每一个样本放在每一列

% Include a row of 1s as an additional intercept feature.
data = [ ones(1,size(data,2)); data ];  % 15x506    添加intercept term 

% Shuffle examples. 乱序 目的在于之后可以随机选取training set和test sets
data = data(:, randperm(size(data,2))); %randperm(n)用于随机生成1到n的排列

% Split into train and test sets
% The last row of 'data' is the median home price.
train.X = data(1:end-1,1:400);   %选择前400个样本来训练，后面的样本来做測试
train.y = data(end,1:400);

test.X = data(1:end-1,401:end);
test.y = data(end,401:end);

m=size(train.X,2);  %训练样本数量
n=size(train.X,1);  %每一个样本的变量个数

% Initialize the coefficient vector theta to random values.
theta = rand(n,1); %随机生成初始theta 每一个值在(0,1)之间

% Run the minFunc optimizer with linear_regression.m as the objective.
%
% TODO:  Implement the linear regression objective and gradient computations
% in linear_regression.m
%
tic; %Start a stopwatch timer. 開始计时
options = struct('MaxIter', 200);
theta = minFunc(@linear_regression, theta, options, train.X, train.y);
fprintf('Optimization took %f seconds.
', toc); %toc Read the stopwatch timer

% Run minFunc with linear_regression_vec.m as the objective.
%
% TODO:  Implement linear regression in linear_regression_vec.m
% using MATLAB's vectorization features to speed up your code.
% Compare the running time for your linear_regression.m and
% linear_regression_vec.m implementations.
%
% Uncomment the lines below to run your vectorized code.
%Re-initialize parameters
%theta = rand(n,1);
%tic;
%theta = minFunc(@linear_regression_vec, theta, options, train.X, train.y);
%fprintf('Optimization took %f seconds.
', toc);

% Plot predicted prices and actual prices from training set.
actual_prices = train.y;
predicted_prices = theta'*train.X;

% Print out root-mean-squared (RMS) training error.平方根误差
train_rms=sqrt(mean((predicted_prices - actual_prices).^2));
fprintf('RMS training error: %f
', train_rms);

% Print out test RMS error
actual_prices = test.y;
predicted_prices = theta'*test.X;
test_rms=sqrt(mean((predicted_prices - actual_prices).^2));
fprintf('RMS testing error: %f
', test_rms);


% Plot predictions on test data.
plot_prices=true;
if (plot_prices)
  [actual_prices,I] = sort(actual_prices); %从小到大排序价格。I为index
  predicted_prices=predicted_prices(I);
  plot(actual_prices, 'rx');
  hold on;
  plot(predicted_prices,'bx');
  legend('Actual Price', 'Predicted Price');
  xlabel('House #');
  ylabel('House price ($1000s)');
end

3.2 linear_regression.m code

function [f,g] = linear_regression(theta, X,y)
  %
  % Arguments:
  %   theta - A vector containing the parameter values to optimize.
  %   X - The examples stored in a matrix.
  %       X(i,j) is the i'th coordinate of the j'th example.
  %   y - The target value for each example.  y(j) is the target for example j.
  %
  
  m=size(X,2);
  n=size(X,1);

  f=0;
  g=zeros(size(theta));

  %
  % TODO:  Compute the linear regression objective by looping over the examples in X.
  %        Store the objective function value in 'f'.
  %
  % TODO:  Compute the gradient of the objective with respect to theta by looping over
  %        the examples in X and adding up the gradient for each example.  Store the
  %        computed gradient in 'g'.
  
%%% YOUR CODE HERE %%%

% Step 1 : Compute f cost function
for i = 1:m
    f = f + (theta' * X(:,i) - y(i))^2;
end

f = 1/2*f;

% Step 2: Compute gradient 

for j = 1:n
    for i = 1:m
        g(j) = g(j) + X(j,i)*(theta' * X(:,i) - y(i));
    end
    
end

3.3 Result

Optimization took 3.374166 seconds.
RMS training error: 4.679871
RMS testing error: 4.865463

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