Distributed Online Optimization With Long-Term Constraints - Yuan, Deming; Proutiere, Alexandre; Shi, Guodong (IEEE transactions on automatic control, 2022)
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.