Distributed Gradient Tracking for Unbalanced Optimization With Different Constraint Sets

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 dire...

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Published inIEEE transactions on automatic control Vol. 68; no. 6; pp. 3633 - 3640
Main Authors Cheng, Songsong, Liang, Shu, Fan, Yuan, Hong, Yiguang
Format Journal Article
LanguageEnglish
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
Online AccessGet full text
ISSN0018-9286
1558-2523
DOI10.1109/TAC.2022.3192316

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Abstract 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>.
AbstractList 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>.
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 [Formula Omitted] convergence rate. Moreover, we design an epoch iteration scheme and improve the convergence rate as [Formula Omitted].
Author Fan, Yuan
Liang, Shu
Hong, Yiguang
Cheng, Songsong
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SubjectTerms 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
Title Distributed Gradient Tracking for Unbalanced Optimization With Different Constraint Sets
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