부산대학교 도서관

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.