more and more control variates

A few months ago, François Portier and Johan Segers arXived a paper on a question that has always puzzled me, namely how to add control variates to a Monte Carlo estimator and when to stop if needed! The paper is called Monte Carlo integration with a growing number of control variates. It is related to the earlier Oates, Girolami and Chopin (2017) which I remember discussing with Chris when he was in Warwick. The puzzling issue of control variates is [for me] that, while the optimal weight always decreases the variance of the resulting estimate, in practical terms, implementing the method may increase the actual variance. Glynn and Szechtman at MCqMC 2000 identify six different ways of creating the estimate, depending on how the covariance matrix, denoted P(hh’), is estimated. With only one version integrating constant functions and control variates exactly. Which actually happens to be also a solution to a empirical likelihood maximisation under the empirical constraints imposed by the control variates. Another interesting feature is that, when the number m of control variates grows with the number n of simulations the asymptotic variance goes to zero, meaning that the control variate estimator converges at a faster speed.

Creating an infinite sequence of control variates sounds unachievable in a realistic situation. Legendre polynomials are used in the paper, but is there a generic and cheap way to getting these. And … control variate selection, anyone?!