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Distributed Online Optimization With Long-Term Constraints.

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Title: Distributed Online Optimization With Long-Term Constraints.
Authors: Yuan, Deming1 dmyuan1012@gmail.com, Proutiere, Alexandre2 alepro@kth.se, Shi, Guodong3 guodong.shi@sydney.edu.au
Source: IEEE Transactions on Automatic Control. Mar2022, Vol. 67 Issue 3, p1089-1104. 16p.
Subjects: Convex functions, Vector valued functions, Time perspective, Distributed algorithms
Abstract: 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. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
Abstract: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. [ABSTRACT FROM AUTHOR]
ISSN:00189286
DOI:10.1109/TAC.2021.3057601