Machine learning（2-Linear regression with one variable ）

1、Model representation

• Our Training Set [训练集]:

• We will start with this ‘’Housing price prediction‘’ example first of fitting linear functions, and we will build on this to eventually have more complex models

2、Cost function

• 代价函数（平方误差函数）：It figures out how to fit the best possible straight line to our data
• So how to choose θ_i's ?

• and just try:

• The parameters we choose determine the accuracy of the straight line we get relative to our training set
• But there is modeling error 建模误差

• Our goal is to select the model parameters that minimize the sum of squares of modeling errors

• That is to minimize the cost function!

• summary：

2-1、Cost function introduction I

• We look up some plots to understand the cost function

2-2、Cost function introduction II

• Let's take a look at the three-dimensional space diagram of the cost function(also called a convex function 凸函数)

• And here is an example of a contour figure:

• The contour figure is a more convenient way to visualize the cost function

3、Gradient descent

• It turns out gradient descent(梯度下降) is a more general algorithm and is used not only in linear regression. I will introduce how to use gradient descent for minimizing some arbitrary function J
• 
• 
• The formula of the batch gradient descent algorithm :

4、Gradient descent intuition

• Derivative term purpose ：get closer to the minimum

• 

• Learning rate α ：

• But what if my parameter θ₁ is already at a local minimum？
• 
• Gradient descent can converge to a local minimum， even with the learning rate α fixed
• 

5、Gradient descent for linear regression

• 

• 

•