Achieving Near-Optimal Convergence for Distributed Minimax Optimization with Adaptive Stepsizes

TL;DR

This paper introduces D-AdaST, a distributed adaptive minimax optimization method with stepsize tracking that guarantees convergence and achieves near-optimal rates, addressing issues of inconsistency in adaptive stepsizes.

Contribution

The paper proposes D-AdaST, the first distributed adaptive minimax method with stepsize tracking that guarantees convergence without prior parameter knowledge.

Findings

Guarantees exact convergence in distributed minimax problems.

Achieves near-optimal convergence rate of (\u03b5^{-(4+)}) for nonconvex-strongly-concave problems.

Validated through extensive experiments.

Abstract

In this paper, we show that applying adaptive methods directly to distributed minimax problems can result in non-convergence due to inconsistency in locally computed adaptive stepsizes. To address this challenge, we propose D-AdaST, a Distributed Adaptive minimax method with Stepsize Tracking. The key strategy is to employ an adaptive stepsize tracking protocol involving the transmission of two extra (scalar) variables. This protocol ensures the consistency among stepsizes of nodes, eliminating the steady-state error due to the lack of coordination of stepsizes among nodes that commonly exists in vanilla distributed adaptive methods, and thus guarantees exact convergence. For nonconvex-strongly-concave distributed minimax problems, we characterize the specific transient times that ensure time-scale separation of stepsizes and quasi-independence of networks, leading to a near-optimal convergence rate of $\tilde{\mathcal{O}} \left( \epsilon ^{-\left( 4+\delta \right)} \right)$ for any small $\delta > 0$, matching that of the centralized counterpart. To our best knowledge, D-AdaST is the first distributed adaptive method achieving near-optimal convergence without knowing any problem-dependent parameters for nonconvex minimax problems. Extensive experiments are conducted to validate our theoretical results.