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Gradient information on the sampling distribution can be used to reduce thevariance of Monte Carlo estimators via Stein's method. An important applicationis that of estimating an expectation of a test function along the sample pathof a Markov chain, where gradient information enables convergence rateimprovement at the cost of a linear system which must be solved. Thecontribution of this paper is to establish theoretical bounds on convergencerates for a class of estimators based on Stein's method. Our analysis accountsfor (i) the degree of smoothness of the sampling distribution and testfunction, (ii) the dimension of the state space, and (iii) the case ofnon-independent samples arising from a Markov chain. These results provideinsight into the rapid convergence of gradient-based estimators observed forlow-dimensional problems, as well as clarifying a curse-of-dimension thatappears inherent to such methods.

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