  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.
