Distributed Saddle-Point Problems Under Similarity

We study solution methods for (strongly-)convex-(strongly)-concave Saddle-Point Problems (SPPs) over networks of two type-master/workers (thus centralized) architectures and mesh (thus decentralized) networks. The local functions at each node are assumed to be similar, due to statistical data similarity or otherwise. We establish lower complexity bounds for a fairly general class of algorithms solving the SPP. We show that a given suboptimality epsilon > 0 is achieved over master/workers networks in Omega(Delta center dot delta/mu center dot log(1/epsilon)) rounds of communications, where delta > 0 measures the degree of similarity of the local functions, mu is their strong convexity constant, and Delta is the diameter of the network. The lower communication complexity bound over mesh networks reads Omega(1/root rho center dot delta/mu center dot log(1/epsilon)), where rho is the (normalized) eigengap of the gossip matrix used for the communication between neighbouring nodes. We then propose algorithms matching the lower bounds over either types of networks (up to log-factors). We assess the effectiveness of the proposed algorithms on a robust regression problem.