Adaptive control variates

Adaptive Monte Carlo methods are specialized Monte Carlo simulation techniques where the methods are adaptively tuned as the simulation progresses. The primary focus of such techniques has been in adaptively tuning importance sampling distributions to reduce the variance of an estimator. We instead focus on adaptive control variate schemes, developing asymptotic theory for the performance of two adaptive control variate estimators. The first estimator is based on a stochastic approximation scheme for identifying the optimal choice of control variate. It is easily implemented, but its performance is sensitive to certain tuning parameters, the selection of which is nontrivial. The second estimator uses a sample average approximation approach. It has the advantage that it does not require any tuning parameters, but it can be computationally expensive and requires the availability of nonlinear optimization software.