Empirical Bernstein and betting confidence intervals for randomized quasi-Monte Carlo

TL;DR

This work provides non-asymptotic confidence intervals for randomized quasi-Monte Carlo estimates when the integrand is bounded, by leveraging empirical Bernstein and hedged betting methods. By modeling the per-replicate variance as $σ_n^2 = σ_0^2 n^{−θ}$, the authors derive that the width-minimizing replicate size is $n = Θ(N^{1/(θ+1)})$, leading to interval widths $Θ(N^{−θ/(θ+1)})$, with stronger RQMC convergence (larger θ) yielding slower growth of the optimal $n$ yet significantly narrower intervals. Finite-sample experiments show hedged betting intervals (HBCI) are typically narrower than empirical Bernstein intervals (EBCI), and that the optimal $n$ grows slowly with budget $N$; in ridge-function tests the dimension has little effect on interval width. The results indicate practical, finite-sample uncertainty quantification for RQMC estimates and highlight the advantage of HBCI in producing tighter, reliable confidence intervals in a range of settings, including those with known variance structure. This advances reliable uncertainty quantification for high-precision numerical integration using RQMC in scientific computing contexts.

Abstract

Randomized quasi-Monte Carlo (RQMC) methods estimate the mean of a random variable by sampling an integrand at $n$ equidistributed points. For scrambled digital nets, the resulting variance is typically $\tilde O(n^{-θ})$ where $θ\in[1,3]$ depends on the smoothness of the integrand and $\tilde O$ neglects logarithmic factors. While RQMC can be far more accurate than plain Monte Carlo (MC) it remains difficult to get confidence intervals on RQMC estimates. We investigate some empirical Bernstein confidence intervals (EBCI) and hedged betting confidence intervals (HBCI), both from Waudby-Smith and Ramdas (2024), when the random variable of interest is subject to known bounds. When there are $N$ integrand evaluations partitioned into $R$ independent replicates of $n=N/R$ RQMC points, and the RQMC variance is $Θ(n^{-θ})$, then an oracle minimizing the width of a Bennett confidence interval would choose $n =Θ(N^{1/(θ+1)})$. The resulting intervals have a width that is $Θ(N^{-θ/(θ+1)})$. Our empirical investigations had optimal values of $n$ grow slowly with $N$, HBCI intervals that were usually narrower than the EBCI ones, and optimal values of $n$ for HBCI that were equal to or smaller than the ones for the oracle.

Figures (2)

• Figure 1: Mean confidence interval width for the HBCI as a function of the RQMC sample size $n$. There is one panel for each of the $4$ ridge functions. Within a panel the curves are for $N=2^{2r}$ for $r=4,5,6,7,8$ from top to bottom. The smallest mean width for each $N$ is marked with a larger circle. The means are taken over all $20$ replicates and all $4$ dimensions used.
• Figure 2: Ratio of mean HBCI width to width from Bennett's inequality as a function of the RQMC sample size $n$ for the given budgets $N$. The means for the HBCI are taken over the 20 independent replicates.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof