Super-polynomial accuracy of multidimensional randomized nets using the median-of-means

Abstract

We study approximate integration of a function $f$ over $[0,1]^s$ based on taking the median of $2r-1$ integral estimates derived from independently randomized $(t,m,s)$-nets in base $2$. The nets are randomized by Matousek's random linear scramble with a digital shift. If $f$ is analytic over $[0,1]^s$, then the probability that any one randomized net's estimate has an error larger than $2^{-cm^2/s}$ times a quantity depending on $f$ is $O(1/\sqrt{m})$ for any $c<3\log(2)/\pi^2\approx 0.21$. As a result the median of the distribution of these scrambled nets has an error that is $O(n^{-c\log(n)/s})$ for $n=2^m$ function evaluations. The sample median of $2r-1$ independent draws attains this rate too, so long as $r/m^2$ is bounded away from zero as $m\to\infty$. We include results for finite precision estimates and some non-asymptotic comparisons to taking the mean of $2r-1$ independent draws.