Efficient Gradient Methods for Distributed Saddle Problems

2026-05-18

AI Summary

This work addresses the lack of theoretical foundations for balancing communication and computation costs in distributed saddle-point problems by proposing a multi-stage residual-norm decoupled minimization gradient algorithm. The method achieves, for the first time, communication-optimal decoupled solutions within a zeroth-order feedback framework and extends this result to generalized variational inequality problems, thereby overcoming the query complexity bottleneck of classical extragradient methods. By integrating multi-stage variance reduction, gradient sliding mechanisms, and a refined communication complexity analysis, the proposed approach attains the current best-known communication complexity for distributed saddle-point problems. Furthermore, it establishes its communication optimality within the class of gradient sliding algorithms through a matching lower bound.

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.

Problem

• Distributed Saddle Problems
• Communication Complexity
• Gradient Methods
• Variational Inequality Problem
• Zero-Sum Games

Innovation

• distributed saddle problems
• communication-optimal
• decoupled method
• variational inequality
• gradient-span algorithms