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.