Decentralized saddle point problems via non-Euclidean mirror prox

Abstract

We consider smooth convex-concave saddle point problems in the decentralized distributed setting, where a finite-sum objective is distributed among the nodes of a computational network. At each node, the local objective depends on the groups of local and global variables. For such problems, we propose a decentralized distributed algorithm with O ( ϵ − 1 ) communication and oracle calls complexities to achieve accuracy ε in terms of the duality gap and in terms of consensus between nodes. Further, we prove lower bounds for the communication and oracle calls complexities and show that our algorithm matches these bounds, i.e. it is optimal. In contrast to existing decentralized algorithms, our algorithm admits non-euclidean proximal setup, including, e.g. entropic. We illustrate the work of the proposed algorithm on the prominent problem of computing Wasserstein barycenters (WB), where a non-euclidean proximal setup arises naturally in a bilinear saddle point reformulation of the WB problem.