Stochastic Fixed-Point Iterations for Nonexpansive Maps: Convergence and Error Bounds

IF 2.2 2区数学 Q2 AUTOMATION & CONTROL SYSTEMS

SIAM Journal on Control and Optimization Pub Date : 2024-01-18 DOI:10.1137/22m1515550

Mario Bravo, Roberto Cominetti

引用次数: 0

Abstract

SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 191-219, February 2024.
Abstract. We study a stochastically perturbed version of the well-known Krasnoselskii–Mann iteration for computing fixed points of nonexpansive maps in finite dimensional normed spaces. We discuss sufficient conditions on the stochastic noise and stepsizes that guarantee almost sure convergence of the iterates towards a fixed point and derive nonasymptotic error bounds and convergence rates for the fixed-point residuals. Our main results concern the case of a martingale difference noise with variances that can possibly grow unbounded. This supports an application to reinforcement learning for average reward Markov decision processes, for which we establish convergence and asymptotic rates. We also analyze in depth the case where the noise has uniformly bounded variance, obtaining error bounds with explicit computable constants.

查看原文本刊更多论文

非扩展映射的随机定点迭代：收敛性与误差边界

SIAM 控制与优化期刊》第 62 卷第 1 期第 191-219 页，2024 年 2 月。摘要。我们研究了著名的 Krasnoselskii-Mann 迭代的随机扰动版本，用于计算有限维规范空间中的无穷映射的定点。我们讨论了随机噪声和步长的充分条件，这些条件保证了迭代几乎肯定收敛于定点，并推导出了非渐近误差边界和定点残差的收敛率。我们的主要结果涉及方差可能无限制增长的鞅差分噪声。这支持了平均报酬马尔可夫决策过程的强化学习应用，我们为其建立了收敛性和渐近率。我们还深入分析了噪声方差均匀有界的情况，获得了具有明确可计算常数的误差边界。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

SIAM Journal on Control and Optimization 数学-应用数学

CiteScore

4.00

自引率

4.50%

发文量

143

审稿时长

12 months

期刊介绍： SIAM Journal on Control and Optimization (SICON) publishes original research articles on the mathematics and applications of control theory and certain parts of optimization theory. Papers considered for publication must be significant at both the mathematical level and the level of applications or potential applications. Papers containing mostly routine mathematics or those with no discernible connection to control and systems theory or optimization will not be considered for publication. From time to time, the journal will also publish authoritative surveys of important subject areas in control theory and optimization whose level of maturity permits a clear and unified exposition. The broad areas mentioned above are intended to encompass a wide range of mathematical techniques and scientific, engineering, economic, and industrial applications. These include stochastic and deterministic methods in control, estimation, and identification of systems; modeling and realization of complex control systems; the numerical analysis and related computational methodology of control processes and allied issues; and the development of mathematical theories and techniques that give new insights into old problems or provide the basis for further progress in control theory and optimization. Within the field of optimization, the journal focuses on the parts that are relevant to dynamic and control systems. Contributions to numerical methodology are also welcome in accordance with these aims, especially as related to large-scale problems and decomposition as well as to fundamental questions of convergence and approximation.