A Discrepancy Bound for a Deterministic Acceptance-Rejection Sampler

TL;DR

This paper establishes a theoretical discrepancy bound for a deterministic acceptance-rejection sampler using quasi-Monte Carlo sequences, demonstrating improved convergence rates over traditional Monte Carlo methods and extending results to general densities via measure-preserving transformations.

Contribution

It provides the first discrepancy bounds for deterministic acceptance-rejection samplers using QMC sequences, including for general densities through the inverse Rosenblatt transformation.

Findings

Discrepancy of samples is bounded by N^{-1/s}.

Lower bounds show discrepancy can be as high as N^{-2/(s+1)}.

Convergence rates improve upon standard Monte Carlo methods.

Abstract

We consider an acceptance-rejection sampler based on a deterministic driver sequence. The deterministic sequence is chosen such that the discrepancy between the empirical target distribution and the target distribution is small. We use quasi-Monte Carlo (QMC) point sets for this purpose. The empirical evidence shows convergence rates beyond the crude Monte Carlo rate of $N^{-1/2}$. We prove that the discrepancy of samples generated by the QMC acceptance-rejection sampler is bounded from above by $N^{-1/s}$. A lower bound shows that for any given driver sequence, there always exists a target density such that the star discrepancy is at most $N^{-2/(s+1)}$. For a general density, whose domain is the real state space $\mathbb{R}^{s-1}$, the inverse Rosenblatt transformation can be used to convert samples from the $(s-1)-$dimensional cube to $\mathbb{R}^{s-1}$. We show that this transformation is measure preserving. This way, under certain conditions, we obtain the same convergence rate for a general target density defined in $\mathbb{R}^{s-1}$. Moreover, we also consider a deterministic reduced acceptance-rejection algorithm recently introduced by Barekat and Caflisch [F. Barekat and R.Caflisch. Simulation with Fluctuation and Singular Rates. ArXiv:1310.4555[math.NA], 2013.]