Distributed Bandit Online Convex Optimization With Time-Varying Coupled Inequality Constraints

• Published in: IEEE transactions on automatic control Vol. 66; no. 10; pp. 4620 - 4635
• Main Authors: Yi, Xinlei, Li, Xiuxian, Yang, Tao, Xie, Lihua, Chai, Tianyou, Johansson, Karl Henrik
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.10.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Approximation algorithms; Bandit convex optimization; Computational geometry; Convex analysis; Convex functions; Convexity; distributed optimization; Electric power grids; Feedback; Games; gradient approximation; Heuristic algorithms; Numerical simulation; online optimization; Optimization; Optimization methods; Tightness; Time factors; time-varying constraints
• DOI: 10.1109/TAC.2020.3030883
• ISSN: 0018-9286; 1558-2523

Distributed bandit online convex optimization with time-varying coupled inequality constraints is considered, motivated by a repeated game between a group of learners and an adversary. The learners attempt to minimize a sequence of global loss functions and at the same time satisfy a sequence of coupled constraint functions, where the constraints are coupled across the distributed learners at each round. The global loss and the coupled constraint functions are the sum of local convex loss and constraint functions, respectively, which are adaptively generated by the adversary. The local loss and constraint functions are revealed in a bandit manner, i.e., only the values of loss and constraint functions are revealed to the learners at the sampling instance, and the revealed function values are held privately by each learner. Both one- and two-point bandit feedback are studied with the two corresponding distributed bandit online algorithms used by the learners. We show that sublinear expected regret and constraint violation are achieved by these two algorithms, if the accumulated variation of the comparator sequence also grows sublinearly. In particular, we show that <inline-formula><tex-math notation="LaTeX">\mathcal {O}(T^{\theta })</tex-math></inline-formula> expected static regret and <inline-formula><tex-math notation="LaTeX">\mathcal {O}(T^{7/4-\theta })</tex-math></inline-formula> constraint violation are achieved in the one-point bandit feedback setting, and <inline-formula><tex-math notation="LaTeX">\mathcal {O}(T^{\max \lbrace \kappa,1-\kappa \rbrace })</tex-math></inline-formula> expected static regret and <inline-formula><tex-math notation="LaTeX">\mathcal {O}(T^{1-\kappa /2})</tex-math></inline-formula> constraint violation in the two-point bandit feedback setting, where <inline-formula><tex-math notation="LaTeX">\theta \in (3/4,5/6]</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">\kappa \in (0,1)</tex-math></inline-formula> are user-defined tradeoff parameters. Finally, the tightness of the theoretical results is illustrated by numerical simulations of a simple power grid example, which also compares the proposed algorithms to algorithms existing in the literature.