Alternating Direction Method of Multipliers for Decomposable Saddle-Point Problems

TL;DR

This paper extends the alternating direction method of multipliers to decomposable saddle-point problems, enabling efficient solutions by exploiting problem structure, with proven convergence under mild conditions and demonstrated effectiveness through numerical examples.

Contribution

It introduces a novel ADMM-based algorithm for decomposable saddle-point problems and establishes its convergence under mild assumptions.

Findings

Convergence of the proposed method is proven for convex-concave saddle-point problems.

The method efficiently solves large-scale decomposable saddle-point problems.

Numerical experiments validate the convergence and practical effectiveness of the algorithm.

Abstract

Saddle-point problems appear in various settings including machine learning, zero-sum stochastic games, and regression problems. We consider decomposable saddle-point problems and study an extension of the alternating direction method of multipliers to such saddle-point problems. Instead of solving the original saddle-point problem directly, this algorithm solves smaller saddle-point problems by exploiting the decomposable structure. We show the convergence of this algorithm for convex-concave saddle-point problems under a mild assumption. We also provide a sufficient condition for which the assumption holds. We demonstrate the convergence properties of the saddle-point alternating direction method of multipliers with numerical examples on a power allocation problem in communication channels and a network routing problem with adversarial costs.