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，出现在许多现实系统中。然而，当嵌套层数较大时，现有方法通常面临高昂的计算成本。固定非负整数 $D$ 为总嵌套层数。标准蒙特卡洛方法通常需要至少 $\mathcal{O}(\varepsilon^{-(2+D)})$ 的代价，有时甚至 $\mathcal{O}(\varepsilon^{-2(1+D)})$，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。本文提出一种名为 $\mathsf{READ}$（“任意深度递归估计器”的缩写）的新型蒙特卡洛估计器。在适当假设下，对于每个固定的 $D$，我们的估计器具有 $\mathcal{O}(\varepsilon^{-2})$ 的最优计算代价；在更一般的假设下，其计算代价近乎最优，为 $\mathcal{O}(\varepsilon^{-2(1 + \delta)})$（其中 $0 < \delta < \frac12$）。该估计器还是无偏的，便于并行化。构造的核心要素在于对问题递归结构的观察，以及随机多层蒙特卡洛方法的递归应用。