Exponential convergence of adaptive importance sampling estimators for Markov chain expectations

TL;DR

This paper proves that adaptive importance sampling algorithms for Markov chain expectations converge exponentially fast under broad conditions, extending previous results and confirming the maximal class of such problems.

Contribution

It establishes exponential convergence of adaptive importance sampling for a wide class of Markov chain expectation problems, expanding prior specialized results.

Findings

Exponential convergence is proven for most combinations of Markov expectations and filtered estimators.

The paper extends known results to a broader class of problems.

It discusses the applicability of Markov chain theory through a counterexample.

Abstract

In this paper it is shown that adaptive importance sampling algorithms converge at exponential rate for Markov chain expectation problems that admit a combination of a filtered estimator and a Markov zero-variance measure. It extends a chain of results---special purpose proofs were already known for several cases (Kollman et al, Baggerly et al, Desai). A recent paper (Awad et al) provides a complete description of the class of combinations of Markov process expectations of path functionals and filtered estimators that admit zero-variance importance measures that retain the Markov property. In a way, this is the maximal class for which adaptive importance sampling algorithms might exhibit exponential convergence. The main purpose of this paper is to prove that this is the case: for (most of) those combinations the natural adaptive importance sampling algorithm converges at exponential rate. In addition, the applicability of general Markov chain theory for this purpose is discussed through the analysis of a counterexample presented in (Desai and Glynn).