线性回归与梯度下降（ML作业）

Loss函数

题目一：完成computeCost.m

function J = computeCost(X, y, theta)
%COMPUTECOST Compute cost for linear regression
%   J = COMPUTECOST(X, y, theta) computes the cost of using theta as the
%   parameter for linear regression to fit the data points in X and y

% Initialize some useful values
m = length(y); % number of training examples

% You need to return the following variables correctly
J = 0;

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta
%               You should set J to the cost.

J = (1 / (2 * m))* sum((X * theta - y).^ 2);

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

end

直接套用公式编写：

\[J = \frac{1}{2m} \sum_{i=0}^{m}(wx - y)^2
\]

Loss函数升级版:computeCostMulti.m

function J = computeCostMulti(X, y, theta)
%COMPUTECOSTMULTI Compute cost for linear regression with multiple variables
%   J = COMPUTECOSTMULTI(X, y, theta) computes the cost of using theta as the
%   parameter for linear regression to fit the data points in X and y

% Initialize some useful values
m = length(y); % number of training examples

% You need to return the following variables correctly
J = 0;

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta
%               You should set J to the cost.

J = (1 / (2 * m))* sum((X * theta - y).^ 2);

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

end

其实写法一模一样……

GD算法：gradientDescent.m

function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters)
%GRADIENTDESCENT Performs gradient descent to learn theta
%   theta = GRADIENTDESCENT(X, y, theta, alpha, num_iters) updates theta by
%   taking num_iters gradient steps with learning rate alpha

% Initialize some useful values
m = length(y); % number of training examples
J_history = zeros(num_iters, 1);

for iter = 1:num_iters

    % ====================== YOUR CODE HERE ======================
    % Instructions: Perform a single gradient step on the parameter vector
    %               theta.
    %
    % Hint: While debugging, it can be useful to print out the values
    %       of the cost function (computeCost) and gradient here.
    %
    temp = zeros(length(theta), 1);
    for i = 1:length(theta)
        temp(i, 1) = theta(i, 1) + alpha * (sum((y - X * theta) .* X(:, i))) / length(X);
    end
    theta = temp;
    % ============================================================

    % Save the cost J in every iteration
    J_history(iter) = computeCost(X, y, theta);

end

end

由于当时写题目的时候就直接按照矩阵的写法写，所以其实复杂版写法也是一样的

GD复杂版算法：gradientDescent.m

function [theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters)
%GRADIENTDESCENTMULTI Performs gradient descent to learn theta
%   theta = GRADIENTDESCENTMULTI(x, y, theta, alpha, num_iters) updates theta by
%   taking num_iters gradient steps with learning rate alpha

% Initialize some useful values
m = length(y); % number of training examples
J_history = zeros(num_iters, 1);

for iter = 1:num_iters

    % ====================== YOUR CODE HERE ======================
    % Instructions: Perform a single gradient step on the parameter vector
    %               theta.
    %
    % Hint: While debugging, it can be useful to print out the values
    %       of the cost function (computeCostMulti) and gradient here.
    %

    temp = zeros(length(theta), 1);
    for i = 1:length(theta)
        temp(i, 1) = theta(i, 1) + alpha * (sum((y - X * theta) .* X(:, i))) / length(X);
    end
    theta = temp;

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

    % Save the cost J in every iteration
    J_history(iter) = computeCostMulti(X, y, theta);

end

end

特征缩放:featureNormalize.m

function [X_norm, mu, sigma] = featureNormalize(X)
%FEATURENORMALIZE Normalizes the features in X
%   FEATURENORMALIZE(X) returns a normalized version of X where
%   the mean value of each feature is 0 and the standard deviation
%   is 1. This is often a good preprocessing step to do when
%   working with learning algorithms.

% You need to set these values correctly
X_norm = X;
mu = zeros(1, size(X, 2));
sigma = zeros(1, size(X, 2));

% ====================== YOUR CODE HERE ======================
% Instructions: First, for each feature dimension, compute the mean
%               of the feature and subtract it from the dataset,
%               storing the mean value in mu. Next, compute the
%               standard deviation of each feature and divide
%               each feature by it's standard deviation, storing
%               the standard deviation in sigma.
%
%               Note that X is a matrix where each column is a
%               feature and each row is an example. You need
%               to perform the normalization separately for
%               each feature.
%
% Hint: You might find the 'mean' and 'std' functions useful.
%
mu = mean(X);
sigma = std(X);

for i = 1:length(X)
    X(i, :) = (X(i, :) - mu) ./ sigma;
end
X_norm = X;

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

end

公式采用：

\[X = \frac{X - \mu}{\sigma} \\
\mu:avg \\
\sigma:std
\]

公式法：normalEqn.m

function [theta] = normalEqn(X, y)
%NORMALEQN Computes the closed-form solution to linear regression
%   NORMALEQN(X,y) computes the closed-form solution to linear
%   regression using the normal equations.

theta = zeros(size(X, 2), 1);

% ====================== YOUR CODE HERE ======================
% Instructions: Complete the code to compute the closed form solution
%               to linear regression and put the result in theta.
%

% ---------------------- Sample Solution ----------------------
theta = (X' * X)^-1 * X' * y;

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

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

end