Achieving Linear Convergence in Distributed Asynchronous Multiagent Optimization

• Published in: IEEE transactions on automatic control Vol. 65; no. 12; pp. 5264 - 5279
• Main Authors: Tian, Ye, Sun, Ying, Scutari, Gesualdo
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.12.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Asynchrony; Convergence; Convex functions; delay; Delays; Directed graphs; Distributed algorithms; distributed optimization; Graph theory; linear convergence; Multiagent systems; nonconvex optimization; Optimization; Robustness (mathematics)
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2020.2977940

This article studies multiagent (convex and nonconvex ) optimization over static digraphs. We propose a general distributed asynchronous algorithmic framework whereby 1) agents can update their local variables as well as communicate with their neighbors at any time, without any form of coordination; and 2) they can perform their local computations using (possibly) delayed, out-of-sync information from the other agents. Delays need not be known to the agent or obey any specific profile, and can also be time-varying (but bounded). The algorithm builds on a tracking mechanism that is robust against asynchrony (in the above sense), whose goal is to estimate locally the average of agents' gradients. When applied to strongly convex functions, we prove that it converges at an R-linear (geometric) rate as long as the step-size is sufficiently small. A sublinear convergence rate is proved, when nonconvex problems and/or diminishing, uncoordinated step-sizes are considered. To the best of our knowledge, this is the first distributed algorithm with provable geometric convergence rate in such a general asynchronous setting. Preliminary numerical results demonstrate the efficacy of the proposed algorithm and validate our theoretical findings.