An efficient primal dual prox method for non-smooth optimization

Abstract

We study the non-smooth optimization problems in machine learning, where both the loss function and the regularizer are non-smooth functions. Previous studies on efficient empirical loss minimization assume either a smooth loss function or a strongly convex regularizer, making them unsuitable for non-smooth optimization. We develop a simple yet efficient method for a family of non-smooth optimization problems where the dual form of the loss function is bilinear in primal and dual variables. We cast a non-smooth optimization problem into a minimax optimization problem, and develop a primal dual prox method that solves the minimax optimization problem at a rate of $$O(1/T)$$O(1/T) assuming that the proximal step can be efficiently solved, significantly faster than a standard subgradient descent method that has an $$O(1/\sqrt{T})$$O(1/T) convergence rate. Our empirical studies verify the efficiency of the proposed method for various non-smooth optimization problems that arise ubiquitously in machine learning by comparing it to the state-of-the-art first order methods.