Efficient Gradient Methods for Distributed Saddle Problems

TL;DR

This paper introduces a new distributed saddle problem method that optimizes communication costs, improves upon existing algorithms, and extends to multiplayer game scenarios, establishing theoretical optimality.

Contribution

It formalizes the distributed saddle problem setup, proposes a decoupled method with optimal communication cost, and extends the approach to multiplayer game settings.

Findings

Achieves optimal communication cost within the zero-respecting framework.

Provides a decoupled method that improves over the Extragradient method.

Extends to multiplayer general-sum games with state-of-the-art communication complexity.

Abstract

The distributed setting for Saddle Problems (SPs) has recently emerged as a framework for various modern applications in machine learning and multiagent systems. Despite its relevance, the theoretical foundations of this setting have not yet been thoroughly established. In this paper, we advance this research direction by formalizing the distributed setup for SPs and providing rigorous definitions of communication and computational costs. Our main result is a novel decoupled method that achieves optimal communication cost within the zero-respecting framework. Our method is based on a multi-stage reduction to the decoupled minimization of residual norms, which yields strict improvements over the best known communication cost for the class and the long-standing oracle cost of the Extragradient method. Further, we show by a matching lower bound that our method is communication-optimal within the family of gradient-span algorithms. Finally, we study the extension of distributed SP into Variational Inequality Problem (VIP), which generalizes two-player zero-sum games to multiplayer general-sum games. We show that our decoupled method achieves a new state-of-the-art communication complexity for this broader class.