The Mirror-Prox Sliding Method for Non-smooth decentralized saddle-point problems

TL;DR

This paper introduces a decentralized sliding method for non-smooth saddle-point problems, achieving near-optimal communication and gradient complexity by generalizing centralized techniques.

Contribution

It extends the sliding method to decentralized non-smooth saddle-point problems using penalization, matching lower bounds for communication and gradient calls.

Findings

Achieves near-optimal communication rounds

Matches lower bounds for gradient calls

Generalizes centralized sliding method to decentralized setting

Abstract

The saddle-point optimization problems have a lot of practical applications. This paper focuses on such non-smooth problems in decentralized case. This work contains generalization of recently proposed sliding for centralized problem. Through specific penalization method and this sliding we obtain algorithm for non-smooth decentralized saddle-point problems. Note, the proposed method approaches lower bounds both for number of communication rounds and calls of (sub-)gradient per node.