On Using Control Variates with Stochastic Approximation for Variational Bayes and its Connection to Stochastic Linear Regression

TL;DR

This paper explores the use of control variates and stochastic linear regression to reduce variance in stochastic approximation methods for variational Bayes, demonstrating improved estimator stability.

Contribution

It derives the optimal control variates for stochastic approximation variational Bayes and connects this approach to stochastic linear regression, showing their equivalence under certain conditions.

Findings

Control variates significantly reduce estimator variance.

The proposed method outperforms existing approaches in variance reduction.

The connection between control variates and stochastic linear regression is established.

Abstract

Recently, we and several other authors have written about the possibilities of using stochastic approximation techniques for fitting variational approximations to intractable Bayesian posterior distributions. Naive implementations of stochastic approximation suffer from high variance in this setting. Several authors have therefore suggested using control variates to reduce this variance, while we have taken a different but analogous approach to reducing the variance which we call stochastic linear regression. In this note we take the former perspective and derive the ideal set of control variates for stochastic approximation variational Bayes under a certain set of assumptions. We then show that using these control variates is closely related to using the stochastic linear regression approximation technique we proposed earlier. A simple example shows that our method for constructing control variates leads to stochastic estimators with much lower variance compared to other approaches.