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Recently, we and several other authors have written about the possibilitiesof using stochastic approximation techniques for fitting variationalapproximations to intractable Bayesian posterior distributions. Naiveimplementations of stochastic approximation suffer from high variance in thissetting. Several authors have therefore suggested using control variates toreduce this variance, while we have taken a different but analogous approach toreducing the variance which we call stochastic linear regression. In this notewe take the former perspective and derive the ideal set of control variates forstochastic approximation variational Bayes under a certain set of assumptions.We then show that using these control variates is closely related to using thestochastic linear regression approximation technique we proposed earlier. Asimple example shows that our method for constructing control variates leads tostochastic estimators with much lower variance compared to other approaches.

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