The Widrow-Hoff Algorithm (Primal Form) Repeat: Until convergence criterion satisfied return: Given a training set and learning rate Initial:  Minimize.

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Presentation transcript:

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. ) (Typo on textbook!)

Gradient and Hessian  Let be a differentiable function. The gradient of functionat a point is defined as  If is a twice differentiable function. The Hessian matrix ofat a point is defined as

Example1:

Example 2: The Hessian is positive semi-definite

Solution of the Least Squares Problem The Normal Equations Notation: Find such that has the smallest value, i.e. This is a quadratic unconstrained minimization problem is the optimal solution if and only if

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

Ridge Regression (Guarantee Exist) where