The expected bit complexity of the von Neumann rejection algorithm

TL;DR

This paper analyzes the bit complexity of von Neumann's rejection algorithm in a perfect random bit model, proposing modifications that achieve optimal expected bit usage for Riemann-integrable densities.

Contribution

It introduces a modified rejection method with an oracle-based quadtree extension that guarantees optimal expected bit complexity for all Riemann-integrable densities.

Findings

The modified algorithm achieves optimal expected bit complexity.

Universal lower bounds are established for the bit complexity.

The approach works for densities on compact sets with an oracle for function bounds.

Abstract

In 1952, von Neumann introduced the rejection method for random variate generation. We revisit this algorithm when we have a source of perfect bits at our disposal. In this random bit model, there are universal lower bounds for generating a random variate with a given density to within an accuracy $\epsilon$ derived by Knuth and Yao, and refined by the authors. In general, von Neumann's method fails in this model. We propose a modification that insures proper behavior for all Riemann-integrable densities on compact sets, and show that the expected number of random bits needed behaves optimally with respect to universal lower bounds. In particular, we introduce the notion of an oracle that evaluates the supremum and infimum of a function on any rectangle of $\mathbb{R}^{d}$, and develop a quadtree-style extension of the classical rejection method.