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.

翻译：重复嵌套期望的估计是一项具有挑战性的任务，广泛出现在许多现实系统之中。然而，当嵌套层数增加时，现有方法通常面临高昂的计算成本。考虑总嵌套层数固定为任意非负整数$D$。标准蒙特卡洛方法通常需要至少$\mathcal{O}(\varepsilon^{-(2+D)})$，有时甚至需要$\mathcal{O}(\varepsilon^{-2(1+D)})$的计算成本才能获得误差为$\varepsilon$的估计量。更先进的方法（如多层蒙特卡洛）目前仅适用于$D = 1$的情况。在本文中，我们提出一种新型蒙特卡洛估计量，称为$\mathsf{READ}$（即“任意深度递归估计器”）。在适当假设下，我们对每个固定$D$实现了$\mathcal{O}(\varepsilon^{-2})$的最优计算成本，并在更一般的假设下，对任意$0 < \delta < \frac12$实现了近乎最优的$\mathcal{O}(\varepsilon^{-2(1 + \delta)})$计算成本。该估计量还具有无偏性，便于并行化。我们构建的关键要素在于对问题递归结构的发现，以及对随机化多层蒙特卡洛方法的递归应用。