Introduction to Convex Optimization for Machine Learning

Introduction to Convex Optimization for Machine Learning John Duchi University of California, Berkeley Practical Machine Learning , Fall 2009. Duchi (UC Berkeley) Convex Optimization for Machine Learning Fall 2009 1 / 53. Outline What is Optimization Convex Sets Convex Functions Convex Optimization Problems Lagrange Duality Optimization Algorithms Take Home Messages Duchi (UC Berkeley) Convex Optimization for Machine Learning Fall 2009 2 / 53. What is Optimization What is Optimization (and why do we care?). Duchi (UC Berkeley) Convex Optimization for Machine Learning Fall 2009 3 / 53. What is Optimization What is Optimization ? Finding the minimizer of a function subject to constraints: minimize f0 (x). x fi (x) 0, i = {1, .. , k}. hj (x) = 0, j = {1, .. , l}. Duchi (UC Berkeley) Convex Optimization for Machine Learning Fall 2009 4 / 53. What is Optimization What is Optimization ? Finding the minimizer of a function subject to constraints: minimize f0 (x).

Convex Optimization Problems Definition An optimization problem is convex if its objective is a convex function, the inequality constraints fj are convex, and the equality constraints hj are affine minimize x f0(x) (Convex function) s.t. fi(x) ≤ 0 (Convex sets) hj(x) = 0 (Affine) Duchi (UC Berkeley) Convex Optimization for Machine Learning ...

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