Gain coefficients for scrambled Halton points

TL;DR

This paper analyzes the variance reduction gains of scrambled Halton points in quasi-Monte Carlo methods, providing explicit bounds that grow logarithmically with dimension, offering a practical alternative to Sobol' and Faure sequences.

Contribution

It establishes explicit bounds on the gain coefficients for scrambled Halton points, showing they grow logarithmically with dimension, unlike Sobol' points which grow exponentially.

Findings

Gain coefficients for scrambled Halton points are bounded by a logarithmic function of dimension.

Scrambled Halton points offer a practical variance reduction in high dimensions.

Bounds are explicit and applicable for dimensions up to one million.

Abstract

Randomized quasi-Monte Carlo, via certain scramblings of digital nets, produces unbiased estimates of $\int_{[0,1]^d}f(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}$ with a variance that is $o(1/n)$ for any $f\in L^2[0,1]^d$. It also satisfies some non-asymptotic bounds where the variance is no larger than some $\Gamma<\infty$ times the ordinary Monte Carlo variance. For scrambled Sobol' points, this quantity $\Gamma$ grows exponentially in $d$. For scrambled Faure points, $\Gamma \leqslant \exp(1)\doteq 2.718$ in any dimension, but those points are awkward to use for large $d$. This paper shows that certain scramblings of Halton sequences have gains below an explicit bound that is $O(\log d)$ but not $O( (\log d)^{1-\epsilon})$ for any $\epsilon>0$ as $d\to\infty$. For $6\leqslant d\leqslant 10^6$, the upper bound on the gain coefficient is never larger than $3/2+\log(d/2)$.