Distributed Gradient Tracking for Unbalanced Optimization With Different Constraint Sets

• Published in: IEEE transactions on automatic control Vol. 68; no. 6; pp. 3633 - 3640
• Main Authors: Cheng, Songsong, Liang, Shu, Fan, Yuan, Hong, Yiguang
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Basic converters; Computational geometry; Constraints; Convergence; Convex functions; Convexity; different constraint sets; Directed graphs; distrib- uted optimization; gradient tracking; Heuristic algorithms; Iterative methods; Linear programming; Multi-agent systems; Optimization; Tracking; unbalanced graphs
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2022.3192316

Gradient tracking methods have become popular for distributed optimization in recent years, partially because they achieve linear convergence using only a constant step-size for strongly convex optimization. In this article, we construct a counterexample on constrained optimization to show that direct extension of gradient tracking by using projections cannot guarantee the correctness. Then, we propose projected gradient tracking algorithms with diminishing step-sizes rather than a constant one for distributed strongly convex optimization with different constraint sets and unbalanced graphs. Our basic algorithm can achieve <inline-formula><tex-math notation="LaTeX">O(\ln T/{T})</tex-math></inline-formula> convergence rate. Moreover, we design an epoch iteration scheme and improve the convergence rate as <inline-formula><tex-math notation="LaTeX">O(1/{T})</tex-math></inline-formula>.