Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication

TL;DR

This paper introduces a new class of distributed continuous-time algorithms for network optimization, analyzing their convergence, privacy, and implementation with discrete-time communication, including event-triggered schemes, under various graph conditions.

Contribution

It develops and analyzes continuous-time distributed algorithms with discrete communication, providing convergence guarantees, privacy insights, and event-triggered communication schemes.

Findings

Exponential convergence under strongly connected, weight-balanced graphs with strongly convex costs.

Asymptotic convergence under connected graphs with convex costs.

Event-triggered schemes that avoid Zeno behavior with convergence guarantees.

Abstract

This paper proposes a novel class of distributed continuous-time coordination algorithms to solve network optimization problems whose cost function is a sum of local cost functions associated to the individual agents. We establish the exponential convergence of the proposed algorithm under (i) strongly connected and weight-balanced digraph topologies when the local costs are strongly convex with globally Lipschitz gradients, and (ii) connected graph topologies when the local costs are strongly convex with locally Lipschitz gradients. When the local cost functions are convex and the global cost function is strictly convex, we establish asymptotic convergence under connected graph topologies. We also characterize the algorithm's correctness under time-varying interaction topologies and study its privacy preservation properties. Motivated by practical considerations, we analyze the algorithm implementation with discrete-time communication. We provide an upper bound on the stepsize that guarantees exponential convergence over connected graphs for implementations with periodic communication. Building on this result, we design a provably-correct centralized event-triggered communication scheme that is free of Zeno behavior. Finally, we develop a distributed, asynchronous event-triggered communication scheme that is also free of Zeno with asymptotic convergence guarantees. Several simulations illustrate our results.