Optimal randomized multilevel Monte Carlo for repeatedly nested expectations

TL;DR

This paper introduces a new Monte Carlo estimator, READ, that efficiently estimates repeatedly nested expectations with optimal or near-optimal computational costs, even at arbitrary nesting depths, improving over existing methods.

Contribution

The paper presents the READ estimator, a novel recursive Monte Carlo method that achieves optimal computational complexity for nested expectations at any depth, extending multilevel Monte Carlo techniques.

Findings

Optimal cost of O(ε^{-2}) for fixed nesting depth D

Nearly optimal cost of O(ε^{-2(1+δ)}) for general assumptions

Unbiased estimator suitable for parallel computation

Abstract

The estimation of repeatedly nested expectations is a challenging task that arises in many real-world systems. However, existing methods generally suffer from high computational costs when the number of nestings becomes large. Fix any non-negative integer $D$ for the total number of nestings. Standard Monte Carlo methods typically cost at least $\mathcal{O}(\varepsilon^{-(2+D)})$ and sometimes $\mathcal{O}(\varepsilon^{-2(1+D)})$ to obtain an estimator up to $\varepsilon$-error. More advanced methods, such as multilevel Monte Carlo, currently only exist for $D = 1$. In this paper, we propose a novel Monte Carlo estimator called $\mathsf{READ}$, which stands for "Recursive Estimator for Arbitrary Depth.'' Our estimator has an optimal computational cost of $\mathcal{O}(\varepsilon^{-2})$ for every fixed $D$ under suitable assumptions, and a nearly optimal computational cost of $\mathcal{O}(\varepsilon^{-2(1 + \delta)})$ for any $0 < \delta < \frac12$ under much more general assumptions. Our estimator is also unbiased, which makes it easy to parallelize. The key ingredients in our construction are an observation of the problem's recursive structure and the recursive use of the randomized multilevel Monte Carlo method.