Probabilistic Contraction Analysis of Iterated Random Operators
This provides a general framework for analyzing convergence in Monte Carlo methods with contractive properties, applicable to problems like continuous state and action Markov decision processes, though it is incremental in extending deterministic contraction analysis to probabilistic settings.
The paper tackles the problem of proving convergence for randomized algorithms that approximate contraction maps using iterated random operators, developing a new stochastic dominance-based proof technique called probabilistic contraction analysis to establish convergence in probability for Markov chains in a limiting regime.
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 paper, 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 paper 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 Markov decision problems as representative applications of the general framework developed here.