Distributed Online Optimization With Long-Term Constraints

• Published in: IEEE transactions on automatic control 2022-03, Vol.67 (3), p.1089-1104
• Main Authors: Yuan, Deming, Proutiere, Alexandre, Shi, Guodong
• Format: Article
• Language: English
• Subjects: Absolute constraint violation; Algorithms; Communication graphs; Computational geometry; Computer networks; Constraint violation; Constraints; Convex functions; Convex optimization; Convexity; Cumulative constraints; Decentralized algorithms; Distributed algorithms; Distributed computer systems; Distributed database systems; Distributed online convex optimization; distributed online convex optimization (DOCO); Distributed systems; Functions; long-term constraints; longterm constraints; one-point bandit feedback; Online convex optimizations; Online optimization; Optimization; regret; Robots; State-of-the-art algorithms; Time-varying systems; Unsolicited e-mail; Upper bound; Upper bounds
• ISSN: 0018-9286; 1558-2523; 1558-2523
• DOI: 10.1109/TAC.2021.3057601

In this article, we consider distributed online convex optimization problems, where the distributed system consists of various computing units connected through a time-varying communication graph. In each time step, each computing unit selects a constrained vector, experiences a loss equal to an arbitrary convex function evaluated at this vector, and may communicate to its neighbors in the graph. The objective is to minimize the system-wide loss accumulated over time. We propose a decentralized algorithm with regret and cumulative constraint violation in {\mathcal O}(T^{\max \lbrace c,1-c\rbrace }) and {\mathcal O}(T^{1-c/2}), respectively, for any c\in (0,1), where T is the time horizon. When the loss functions are strongly convex, we establish improved regret and constraint violation upper bounds in {\mathcal O}(\log (T)) and {\mathcal O}(\sqrt{T\log (T)}). These regret scalings match those obtained by state-of-the-art algorithms and fundamental limits in the corresponding centralized online optimization problem (for both convex and strongly convex loss functions). In the case of bandit feedback, the proposed algorithms achieve a regret and constraint violation in {\mathcal O}(T^{\max \lbrace c,1-c/3 \rbrace }) and {\mathcal O}(T^{1-c/2}) for any c\in (0,1). We numerically illustrate the performance of our algorithms for the particular case of distributed online regularized linear regression problems on synthetic and real data.