m=number of training examples
n=number of features
x="input" variables / features
y="output"variable/"target" variable
\((x^{(i)}，y^{(i)})\) = the ith trainging example
\(h_\theta\) = fitting function
其中\(h_\theta(x)=\theta_0+\theta_1x_1+...+\theta_nx_n=\sum_{i=0}^{n}{\theta_ix_i}=\theta^T\)
假设损失函数为\(J(\theta)=\frac{1}{2}\sum_{i=1}^{m}{(h_\theta(x)-y)^2}\) ， To minimize the \(J(\theta)\)
main idea: Initalize \(\theta\) (may \(\theta=\vec{0}\)) ,then keep changing \(\theta\) to reduce \(J(\theta)\) ,untill minimum
Gradient decent:
只有一个样本时,对第i个参数进行更新 \(\theta_i:=\theta_i-\alpha\frac{\partial }{\partial \theta_i}J(\theta)=\theta_i-\alpha(h_\theta(x)-y)x_i\)
Repeat until convergence(收敛):
{
\(\theta_i:=\theta_i-\alpha\sum_{j=1}^{m}(h_\theta(x^{(j)})-y^{j})x_i^{(j)}\) ，(for every i)
}
矩阵描述（简单）：
Repeat until convergence(收敛):
{
\(\theta:=\theta -\nabla_\theta J\)
}
IF \(A\epsilon R^{n*n}\)
? tr(A)=\(\sum_{i=1}^nA_{ii}\) :A的迹
\(J(\theta)=\frac{1}{2}(X\theta - \vec{y})^T(X\theta - \vec{y})\)
\(\nabla_\theta J=\frac{1}{2}\nabla_\theta (\theta^TX^TX\theta-\theta^TX^Ty-y^Tx\theta+y^Ty) =X^TX\theta-X^Ty\)
备注：
当目标函数是凸函数时，梯度下降法的解才是全局最优解
Repeat until convergence:
{
? for j=1 to m{
? \(\theta_i:=\theta_i-\alpha(h_\theta(x^{(j)})-y^{j})x_i^{(j)}\) ，for every i
? }
}
备注：
1.训练速度很快，每次仅仅采用一个样本来迭代；
2.解可能不是最优解，仅仅用一个样本决定梯度方向；
3.不能很快收敛，迭代方向变化很大。
Repeat until convergence:
{
? for j=1 to m/n{
? \(\theta_i:=\theta_i-\alpha\sum_{j=1}^{n}(h_\theta(x^{(j)})-y^{j})x_i^{(j)}\) ，for every i
? }
}
备注：
机器学习中往往采用该算法
参考地址：
https://www.cnblogs.com/pinard/p/5970503.html
原文：https://www.cnblogs.com/daizigege/p/12213522.html