1. Linear Regression  Using a linear function to interpolate the training set  The most popular criterion: Least squares approach  Given the training set: Find a linear function: where is determined by solving the minimization problem:  The function is called the square loss function

2. Linear Regression (Cont.)  Different measures of loss are possible  1-norm loss function  -insensitive loss function  Huber’s regression  Ridge regression where

3. Solution of the Least Squares Problem Some notations: We are going to find the with the samllest square loss. i.e.,

4. The Widrow-Hoff Algorithm (Primal Form) Repeat: Until convergence criterion satisfied return: Given a training set and learning rate Initial:  Minimize the square loss function using gradient descent  Dual form exists (i.e. )

5. The Normal Equations of LSQ Letting we have the normal equations of LSQ: If is inversable then Note: The above result is based on the First Order Optimality Conditions (necessary & sufficient for differentiable convex minimization problems) is singular ? What if