Dimension-independent convergence rates of randomized nets using median-of-means

TL;DR

This paper proves that median-of-means estimators using randomized digital nets achieve dimension-independent convergence rates for high-dimensional integrals, under conditions of low effective dimensionality, improving upon worst-case bounds.

Contribution

It establishes the first dimension-independent convergence rates for median-of-means estimators with randomized digital nets under realistic, weaker assumptions.

Findings

Median-of-means estimators achieve dimension-independent convergence.

Convergence rates are faster under low effective dimensionality.

Results hold under probabilistic, integrand-specific error criteria.

Abstract

Recent advances in quasi-Monte Carlo integration demonstrate that the median of linearly scrambled digital net estimators achieves near-optimal convergence rates for high-dimensional integrals without requiring a priori knowledge of the integrand's smoothness. Building on this framework, we prove that the median estimator attains dimension-independent convergence, a property known as strong tractability in complexity theory, under tractability conditions characterized by low effective dimensionality. Using a probabilistic, integrand-specific error criterion, our analysis establishes both faster and dimension-independent convergence under weaker assumptions than previously possible in the worst-case setting.