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

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 $\tilde O(n^{-\theta})$ where $\theta\in[1,3]$ 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 $\Theta(n^{-\theta})$, then an oracle minimizing the width of a Bennett confidence interval would choose $n =\Theta(N^{1/(\theta+1)})$. The resulting intervals have a width that is $\Theta(N^{-\theta/(\theta+1)})$. 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.