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This paper studies the use of a machine learning-based estimator as a controlvariate for mitigating the variance of Monte Carlo sampling. Specifically, weseek to uncover the key factors that influence the efficiency of controlvariates in reducing variance. We examine a prototype estimation problem thatinvolves simulating the moments of a Sobolev function based on observationsobtained from (random) quadrature nodes. Firstly, we establish aninformation-theoretic lower bound for the problem. We then study a specificquadrature rule that employs a nonparametric regression-adjusted controlvariate to reduce the variance of the Monte Carlo simulation. We demonstratethat this kind of quadrature rule can improve the Monte Carlo rate and achievethe minimax optimal rate under a sufficient smoothness assumption. Due to theSobolev Embedding Theorem, the sufficient smoothness assumption eliminates theexistence of rare and extreme events. Finally, we show that, in the presence ofrare and extreme events, a truncated version of the Monte Carlo algorithm canachieve the minimax optimal rate while the control variate cannot improve theconvergence rate.

When can Regression-Adjusted Control Variates Help? Rare Events, Sobolev Embedding and Minimax Optimality