Decentralized Min-Max Optimization with Gradient Tracking

TL;DR

This paper introduces distributed gradient tracking methods for min-max optimization, enabling flexible multi-agent coordination with theoretical guarantees and superior empirical performance.

Contribution

It proposes two novel distributed gradient methods, DGTA and DSGTA, with matching centralized iteration and sample complexities for nonconvex strongly concave problems.

Findings

DGTA achieves $ ext{O}( ext{kappa}^2 ext{epsilon}^{-2})$ iteration complexity.

DSGTA attains $ ext{O}( ext{kappa}^3 ext{epsilon}^{-4})$ sample complexity.

Numerical experiments show superior empirical performance.

Abstract

This paper presents a novel distributed formulation of the min-max optimization problem. Such a formulation enables enhanced flexibility among agents when optimizing their maximization variables. To address the problem, we propose two distributed gradient methods over networks, termed Distributed Gradient Tracking Ascent (DGTA) and Distributed Stochastic Gradient Tracking Ascent (DSGTA). We demonstrate that DGTA achieves an iteration complexity of $\mathcal{O}(\kappa^2\varepsilon^{-2})$, and DSGTA attains a sample complexity of $\mathcal{O}(\kappa^3\varepsilon^{-4})$ for nonconvex strongly concave (NC-SC) objective functions. Both results match those of their centralized counterparts up to constant factors related to the communication network. Numerical experiments further demonstrate the superior empirical performance of the proposed algorithms compared to existing methods.