On dropping the first Sobol' point

TL;DR

This paper investigates the impact of skipping the first point in Sobol' sequences used in Quasi-Monte Carlo methods, revealing that such practice can significantly increase numerical error and reduce integration accuracy.

Contribution

The study provides a detailed analysis of how skipping the first Sobol' point affects QMC performance, highlighting potential pitfalls of common practices.

Findings

Skipping the first Sobol' point can increase error by a factor proportional to √n.

Retained points often do not form a digital net, affecting error bounds.

Common practices like thinning or burn-in can harm QMC accuracy.

Abstract

Quasi-Monte Carlo (QMC) points are a substitute for plain Monte Carlo (MC) points that greatly improve integration accuracy under mild assumptions on the problem. Because QMC can give errors that are $o(1/n)$ as $n\to\infty$, changing even one point can change the estimate by an amount much larger than the error would have been and worsen the convergence rate. As a result, certain practices that fit quite naturally and intuitively with MC points are very detrimental to QMC performance. These include thinning, burn-in, and taking sample sizes such as powers of $10$, other than the ones for which the QMC points were designed. This article looks at the effects of a common practice in which one skips the first point of a Sobol' sequence. The retained points ordinarily fail to be a digital net and when scrambling is applied, skipping over the first point can increase the numerical error by a factor proportional to $\sqrt{n}$ where $n$ is the number of function evaluations used.