Scalable Control Variates for Monte Carlo Methods via Stochastic Optimization

TL;DR

This paper introduces a scalable framework for control variates in Monte Carlo methods using stochastic optimization, improving variance reduction in large-scale, high-dimensional problems with theoretical insights and empirical validation.

Contribution

It generalizes existing control variate approaches with Stein operators and proposes a stochastic optimization strategy for scalable variance reduction.

Findings

Effective variance reduction demonstrated in high-dimensional settings

Theoretical bounds on variance reduction achieved

Empirical success in Bayesian inference applications

Abstract

Control variates are a well-established tool to reduce the variance of Monte Carlo estimators. However, for large-scale problems including high-dimensional and large-sample settings, their advantages can be outweighed by a substantial computational cost. This paper considers control variates based on Stein operators, presenting a framework that encompasses and generalizes existing approaches that use polynomials, kernels and neural networks. A learning strategy based on minimising a variational objective through stochastic optimization is proposed, leading to scalable and effective control variates. Novel theoretical results are presented to provide insight into the variance reduction that can be achieved, and an empirical assessment, including applications to Bayesian inference, is provided in support.