This paper studies the use of a machine learning-based estimator as a control variate for mitigating the variance of Monte Carlo sampling. Specifically, we seek to uncover the key factors that influence the efficiency of control variates in reducing variance. We examine a prototype estimation problem that involves simulating the moments of a Sobolev function based on observations obtained from (random) quadrature nodes. Firstly, we establish an information-theoretic lower bound for the problem. We then study a specific quadrature rule that employs a nonparametric regression-adjusted control variate to reduce the variance of the Monte Carlo simulation. We demonstrate that this kind of quadrature rule can improve the Monte Carlo rate and achieve the minimax optimal rate under a sufficient smoothness assumption. Due to the Sobolev Embedding Theorem, the sufficient smoothness assumption eliminates the existence of rare and extreme events. Finally, we show that, in the presence of rare and extreme events, a truncated version of the Monte Carlo algorithm can achieve the minimax optimal rate while the control variate cannot improve the convergence rate.

翻译：本文研究以基于机器学习的估计量作为控制变量来降低蒙特卡罗采样方差的问题。具体而言，我们致力于揭示影响控制变量降方差效率的关键因素。我们考察一个原型估计问题：基于（随机）求交节点获取的观测值，仿真索伯列夫函数的矩。首先，我们建立该问题的信息论下界。随后研究一种特定的求交规则，该规则采用非参数回归调整控制变量来降低蒙特卡罗模拟的方差。我们证明，在充分光滑性假设下，此类求交规则能够改进蒙特卡罗速率并达到极小极大最优速率。根据索伯列夫嵌入定理，充分光滑性假设排除了稀有极端事件的存在。最后，我们证明在存在稀有极端事件时，截断版蒙特卡罗算法能够达到极小极大最优速率，而控制变量则无法改善收敛速率。