Automatic optimal-rate convergence of randomized nets using median-of-means

TL;DR

This paper demonstrates that using the median of independently generated randomized digital net estimators achieves near-optimal convergence rates for integral approximation, without prior knowledge of function spaces, simplifying implementation and improving robustness.

Contribution

It introduces a median-based estimator for randomized quasi-Monte Carlo methods that attains optimal convergence rates without requiring pre-designed digital nets or prior function space knowledge.

Findings

Median of estimators approximates integrals at near-optimal rates

Method is easier to implement than previous approaches

Median filtering reduces heavy-tailed errors in smooth integrands

Abstract

We study the sample median of independently generated quasi-Monte Carlo estimators based on randomized digital nets and prove it approximates the target integral value at almost the optimal convergence rate for various function spaces. In contrast to previous methods, the algorithm does not require a priori knowledge of underlying function spaces or even an input of pre-designed $(t,m,s)$-digital nets, and is therefore easier to implement. This study provides further evidence that quasi-Monte Carlo estimators are heavy-tailed when applied to smooth integrands and taking the median can significantly improve the error by filtering out the outliers.