Meta-learning Control Variates: Variance Reduction with Limited Data

Abstract

Control variates can be a powerful tool to reduce the variance of Monte Carlo estimators, but constructing effective control variates can be challenging when the number of samples is small. In this paper, we show that when a large number of related integrals need to be computed, it is possible to leverage the similarity between these integration tasks to improve performance even when the number of samples per task is very small. Our approach, called meta learning CVs (Meta-CVs), can be used for up to hundreds or thousands of tasks. Our empirical assessment indicates that Meta-CVs can lead to significant variance reduction in such settings, and our theoretical analysis establishes general conditions under which Meta-CVs can be successfully trained.

Regularized Zero-Variance Control Variates

“…Theoretical properties of the methods are described, with further details available in Table 1 of South et al (2022b). In addition to these choices of H, choices based on neural networks have been considered (Sun et al, 2023;Wan et al, 2020).…”

“…This is computationally convenient for some optimisation methods, for example least squares, but none of the methods exploit the possible correlation between the r functions to improve the coefficient estimates. Sun et al (2023) use ideas from transfer learning to estimate control variates for multiple functions simultaneously. Their approach is an extension to CF, but one could also consider extending parametric approaches by borrowing ideas from multi-task regression.…”