Unbiased estimators and multilevel Monte Carlo

TL;DR

This paper explores the connection between multilevel Monte Carlo and unbiased estimators, introducing a new class of estimators that improve variance reduction and efficiency, matching MLMC's performance under certain conditions.

Contribution

It presents a new general class of unbiased estimators that unify previous schemes and introduces stratified variants with lower variance, achieving asymptotic efficiency similar to MLMC.

Findings

New estimators achieve substantial variance reduction.

Proposed schemes are asymptotically as efficient as MLMC.

Experiments confirm improved variance reduction.

Abstract

Multilevel Monte Carlo (MLMC) and unbiased estimators recently proposed by McLeish (Monte Carlo Methods Appl., 2011) and Rhee and Glynn (Oper. Res., 2015) are closely related. This connection is elaborated by presenting a new general class of unbiased estimators, which admits previous debiasing schemes as special cases. New lower variance estimators are proposed, which are stratified versions of earlier unbiased schemes. Under general conditions, essentially when MLMC admits the canonical square root Monte Carlo error rate, the proposed new schemes are shown to be asymptotically as efficient as MLMC, both in terms of variance and cost. The experiments demonstrate that the variance reduction provided by the new schemes can be substantial.