Probabilistic Contraction Analysis of Iterated Random Operators

• 成果类型: Article
• 署名作者: Gupta, Abhishek; Jain, Rahul; Glynn, Peter
• 署名单位: University System of Ohio; Ohio State University; University of Southern California; University of Southern California; Stanford University; Stanford University
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2024.3362686
• 发表日期: 2024
• 页码: 5947-5962
• 关键词: Probabilistic logic; Markov processes; Approximation algorithms; CONVERGENCE; COSTS; Extraterrestrial measurements; Monte Carlo methods; Convergence of numerical methods; Iterative algorithms; optimization

摘要

In many branches of engineering, Banach contraction mapping theorem is employed to establish the convergence of certain deterministic algorithms. Randomized versions of these algorithms have been developed that have proved useful in data-driven problems. In a class of randomized algorithms, in each iteration, the contraction map is approximated with an operator that uses independent and identically distributed samples of certain random variables. This leads to iterated random operators acting on an initial point in a complete metric space, and it generates a Markov chain. In this article, we develop a new stochastic dominance-based proof technique, called probabilistic contraction analysis, for establishing the convergence in probability of Markov chains generated by such iterated random operators in certain limiting regime. The methods developed in this article provides a general framework for understanding convergence of a wide variety of Monte Carlo methods, in which contractive property is present. We apply the convergence result to conclude the convergence of fitted value iteration and fitted relative value iteration in continuous state and continuous action MDPs as representative applications of the general framework developed here.