# Control variate selection for Monte Carlo integration

## TL;DR

This paper explores the use of regularized control variate selection via Lasso in Monte Carlo integration, improving efficiency and stability when using many control variates.

## Contribution

It introduces a Lasso-based regularization method for selecting control variates in Monte Carlo integration, enhancing accuracy and computational stability.

## Key findings

- Lasso regularization improves estimator stability.
- Using many control variates increases efficiency.
- Concentration inequalities confirm numerical results.

## Abstract

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 with the integrand as response and the control variates as covariates. Even without special knowledge on the integrand, significant efficiency gains can be obtained if the control variate space is sufficiently large. Incorporating a large number of control variates in the ordinary least squares procedure may however result in (i) a certain instability of the ordinary least squares estimator and (ii) a possibly prohibitive computation time. Regularizing the ordinary least squares estimator by preselecting appropriate control variates via the Lasso turns out to increase the accuracy without additional computational cost. The findings in the numerical experiment are confirmed by concentration inequalities for the integration error.