Mirror Prox Algorithm for Multi-Term Composite Minimization and Semi-Separable Problems

TL;DR

This paper introduces a composite Mirror Prox algorithm for convex-concave saddle point problems, achieving optimal efficiency and applicability to complex regularized problems like Lasso with multiple penalties.

Contribution

It develops a novel composite version of the Mirror Prox algorithm that handles structured saddle point problems efficiently and extends to multi-penalty regularized optimization.

Findings

Achieves $O(1/\epsilon)$ convergence rate for composite saddle point problems.

Applicable to Lasso-type problems with multiple regularizations.

Extends to problems similar to alternating direction methods.

Abstract

In the paper, we develop a composite version of Mirror Prox algorithm for solving convex-concave saddle point problems and monotone variational inequalities of special structure, allowing to cover saddle point/variational analogies of what is usually called "composite minimization" (minimizing a sum of an easy-to-handle nonsmooth and a general-type smooth convex functions "as if" there were no nonsmooth component at all). We demonstrate that the composite Mirror Prox inherits the favourable (and unimprovable already in the large-scale bilinear saddle point case) $O(1/\epsilon)$ efficiency estimate of its prototype. We demonstrate that the proposed approach can be naturally applied to Lasso-type problems with several penalizing terms (e.g. acting together $\ell_1$ and nuclear norm regularization) and to problems of the structure considered in the alternating directions methods, implying in both cases methods with the $O(\epsilon^{-1})$ complexity bounds.