用于重复嵌套期望的最优随机化多层蒙特卡洛方法
Optimal randomized multilevel Monte Carlo for repeatedly nested expectations
AI总结：
本文针对重复嵌套期望估计计算成本随嵌套深度急剧增长的问题，提出递归估计量READ，利用问题递归结构和随机化多层蒙特卡洛方法，在固定深度下实现最优计算成本。
摘要：
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，当嵌套数量变大时，现有方法通常面临高昂的计算成本。固定任意非负整数 $D$ 表示嵌套总数。标准蒙特卡洛方法通常至少需要 $\mathcal{O}(\varepsilon^{-(2+D)})$ 的计算成本，有时甚至需要 $\mathcal{O}(\varepsilon^{-2(1+D)})$ 才能获得误差不超过 $\varepsilon$ 的估计量。更先进的方法，如多层蒙特卡洛，目前仅存在于 $D = 1$ 的情形。在本文中，我们提出了一种新颖的蒙特卡洛估计量，称为 $\mathsf{READ}$，即“任意深度递归估计量”（Recursive Estimator for Arbitrary Depth）。在适当假设下，对于每个固定的 $D$，我们的估计量具有 $\mathcal{O}(\varepsilon^{-2})$ 的最优计算成本；在更一般的假设下，对于任意 $0 < \delta < \frac12$，具有 $\mathcal{O}(\varepsilon^{-2(1+\delta)})$ 的近最优计算成本。我们的估计量还是无偏的，这使得它易于并行化。我们构造中的关键要素是对问题递归结构的观察以及对随机化多层蒙特卡洛方法的递归使用。
英文摘要：
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 $\mathcal{O}(\varepsilon^{-2(1 + δ)})$ for any $0 < δ< \frac12$ 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.
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