Asymptotic Properties of S-AB Method With Diminishing Step-Size

• 成果类型: Article
• 署名作者: Zhao, Shengchao; Liu, Yongchao
• 署名单位: Dalian University of Technology
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2023.3319155
• 发表日期: 2024
• 页码: 3222-3229
• 关键词: convergence; optimization; Covariance matrices; Stochastic processes; linear programming; Gaussian distribution; Approximation algorithms; S-AB; Asymptotic Normality; convergence rate; distributed stochastic optimization (DSO)

摘要

The popular AB /push-pull method for distributed optimization problem may unify much of the existing decentralized first-order methods based on gradient tracking technique. More recently, the stochastic gradient variant of AB /push-pull method ( S - AB ) has been proposed, which achieves the linear rate of converging to a neighborhood of the global minimizer when the step-size is constant. This article is devoted to the asymptotic properties of S - AB with diminishing step-size. Specifically, under the condition that each local objective is smooth and the global objective is strongly convex, we first present the boundedness of the iterates of S - AB and then show that the iterates converge to the global minimizer with the rate O(1/k) in the mean square sense. Furthermore, the asymptotic normality of Polyak-Ruppert averaged S - AB is obtained and applications on statistical inference are discussed. Finally, numerical tests are conducted to demonstrate the theoretic results.