Convergence Rates for a Class of Estimators Based on Stein's Identity

Gradient information on the sampling distribution can be used to reduce Monte Carlo variance. An important application is that of estimating an expectation along the sample path of a Markov chain, where empirical results indicate that gradient information enables improvement on root-$n$ convergence. The contribution of this paper is to establish theoretical bounds on convergence rates for estimators that exploit gradient information, specifically, for a class of estimator based on Stein's identity. Our analysis is refined and accounts for (i) the degree of smoothness of the sampling distribution and test function, (ii) the dimension of the state space, and (iii) the case of non-independent samples arising from a Markov chain. These results provide much-needed insight into the rapid convergence of gradient-based estimators observed for low-dimensional problems, as well as clarifying a curse-of-dimensionality that appears inherent to such methods without further modification.