Ensemble Control Variates

Abstract

Control variates have become an increasingly popular variance-reduction technique in Bayesian inference. Many broadly applicable control variates are based on the Langevin-Stein operator, which leverages gradient information from any gradient-based sampler to produce variance-reduced estimators of expectations. These control variates typically require optimising over a function $u(\theta)$ within a user-defined functional class $G$, such as the space of $Q$th-order polynomials or a reproducing kernel Hilbert space. We propose using averaging-based ensemble learning to construct Stein-based control variates. While the proposed framework is broadly applicable, we focus on ensembles constructed from zero-variance control variates (ZVCV), a popular parametric approach based on solving a linear approximation problem that can easily be over-parameterised in medium-to-high dimensional settings. A common remedy is to use regularised ZVCV via penalised regression, but these methods can be prohibitively slow. We introduce ensemble ZVCV methods based on ensembles of OLS estimators and evaluate the proposed methods against established methods in the literature in a simulation study. Our results show that ensemble ZVCV methods are competitive with regularised ZVCV methods in terms of statistical efficiency, but are substantially faster. This work opens a new direction for constructing broadly applicable control variate techniques via ensemble learning.