Distributed Online Optimization With Long-Term Constraints

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 arb...

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Published in:	IEEE transactions on automatic control Vol. 67; no. 3; pp. 1089 - 1104
Main Authors:	Yuan, Deming, Proutiere, Alexandre, Shi, Guodong
Format:	Journal Article
Language:	English
Published:	New York IEEE 01.03.2022 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
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
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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 <inline-formula><tex-math notation="LaTeX">{\mathcal O}(T^{\max \lbrace c,1-c\rbrace })</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">{\mathcal O}(T^{1-c/2})</tex-math></inline-formula>, respectively, for any <inline-formula><tex-math notation="LaTeX">c\in (0,1)</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">T</tex-math></inline-formula> is the time horizon. When the loss functions are strongly convex, we establish improved regret and constraint violation upper bounds in <inline-formula><tex-math notation="LaTeX">{\mathcal O}(\log (T))</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">{\mathcal O}(\sqrt{T\log (T)})</tex-math></inline-formula>. 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 <inline-formula><tex-math notation="LaTeX">{\mathcal O}(T^{\max \lbrace c,1-c/3 \rbrace })</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">{\mathcal O}(T^{1-c/2})</tex-math></inline-formula> for any <inline-formula><tex-math notation="LaTeX">c\in (0,1)</tex-math></inline-formula>. We numerically illustrate the performance of our algorithms for the particular case of distributed online regularized linear regression problems on synthetic and real data.
AbstractList	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 O(T c,1-c ) and O(T1-c/2), respectively, for any cin (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 O( (T)) and O(T (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 O(T c,1-c/3 ) and O(T1-c/2) for any cin (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. 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 <inline-formula><tex-math notation="LaTeX">{\mathcal O}(T^{\max \lbrace c,1-c\rbrace })</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">{\mathcal O}(T^{1-c/2})</tex-math></inline-formula>, respectively, for any <inline-formula><tex-math notation="LaTeX">c\in (0,1)</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">T</tex-math></inline-formula> is the time horizon. When the loss functions are strongly convex, we establish improved regret and constraint violation upper bounds in <inline-formula><tex-math notation="LaTeX">{\mathcal O}(\log (T))</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">{\mathcal O}(\sqrt{T\log (T)})</tex-math></inline-formula>. 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 <inline-formula><tex-math notation="LaTeX">{\mathcal O}(T^{\max \lbrace c,1-c/3 \rbrace })</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">{\mathcal O}(T^{1-c/2})</tex-math></inline-formula> for any <inline-formula><tex-math notation="LaTeX">c\in (0,1)</tex-math></inline-formula>. We numerically illustrate the performance of our algorithms for the particular case of distributed online regularized linear regression problems on synthetic and real data. 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 [Formula Omitted] and [Formula Omitted], respectively, for any [Formula Omitted], where [Formula Omitted] is the time horizon. When the loss functions are strongly convex, we establish improved regret and constraint violation upper bounds in [Formula Omitted] and [Formula Omitted]. 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 [Formula Omitted] and [Formula Omitted] for any [Formula Omitted]. We numerically illustrate the performance of our algorithms for the particular case of distributed online regularized linear regression problems on synthetic and real data.
Author	Yuan, Deming Proutiere, Alexandre Shi, Guodong
Author_xml	– sequence: 1 givenname: Deming orcidid: 0000-0003-4371-5105 surname: Yuan fullname: Yuan, Deming email: dmyuan1012@gmail.com organization: School of Automation, Nanjing University of Science and Technology, Nanjing, China – sequence: 2 givenname: Alexandre orcidid: 0000-0002-4679-4673 surname: Proutiere fullname: Proutiere, Alexandre email: alepro@kth.se organization: Department of Automatic Control, KTH Royal Institute of Technology, Stockholm, Sweden – sequence: 3 givenname: Guodong orcidid: 0000-0002-5929-9655 surname: Shi fullname: Shi, Guodong email: guodong.shi@sydney.edu.au organization: Australian Center for Field Robotics, Sydney Institute for Robotics and Intelligent Systems, The University of Sydney, Sydney, NSW, Australia
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SubjectTerms	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
Title	Distributed Online Optimization With Long-Term Constraints
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