  Zero-variance control variates (ZV-CV) are a post-processing method to reduce
the variance of Monte Carlo estimators of expectations using the derivatives of
the log target. Once the derivatives are available, the only additional
computational effort lies in solving a linear regression problem. Significant
variance reductions have been achieved with this method in low dimensional
examples, but the number of covariates in the regression rapidly increases with
the dimension of the target. In this paper, we present compelling empirical
evidence that the use of penalized regression techniques in the selection of
high-dimensional control variates provides performance gains over the classical
least squares method. Another type of regularization based on using subsets of
derivatives, or a priori regularization as we refer to it in this paper, is
also proposed to reduce computational and storage requirements. Several
examples showing the utility and limitations of regularized ZV-CV for Bayesian
inference are given. The methods proposed in this paper are accessible through
the R package ZVCV.
