Monte Carlo integration with a growing number of control variates

TL;DR

This paper analyzes Monte Carlo integration using an increasing number of control variates, establishing a central limit theorem and showing faster convergence rates when the control variates densely span the function space.

Contribution

It introduces a central limit theorem for Monte Carlo integration with many control variates and demonstrates conditions for faster convergence rates.

Findings

Central limit theorem for large control variates

Faster convergence when control variates span the function space

Trade-offs between control variate complexity and sample size

Abstract

It is well known that Monte Carlo integration with variance reduction by means of control variates can be implemented by the ordinary least squares estimator for the intercept in a multiple linear regression model. A central limit theorem is established for the integration error if the number of control variates tends to infinity. The integration error is scaled by the standard deviation of the error term in the regression model. If the linear span of the control variates is dense in a function space that contains the integrand, the integration error tends to zero at a rate which is faster than the square root of the number of Monte Carlo replicates. Depending on the situation, increasing the number of control variates may or may not be computationally more efficient than increasing the Monte Carlo sample size.