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

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.

💡 Research Summary

The paper investigates the convergence behavior of randomized quasi‑Monte Carlo (RQMC) methods that employ linearly scrambled base‑2 digital nets, focusing on the median‑of‑means (also called the median trick) estimator. Classical QMC replaces random sampling with low‑discrepancy point sets, achieving faster convergence than Monte Carlo for high‑dimensional integrals. When digital nets are randomized by linear scrambling, the mean‑squared error (MSE) can improve to nearly O(n⁻³ᐟ²) under smoothness assumptions, but outliers in individual replicates often prevent the estimator from attaining this rate when simple averaging is used.

To overcome this limitation, the authors adopt the median‑of‑means approach: they generate 2r − 1 independent replicates of the scrambled net estimator and take the sample median as the final estimate. Lemma 2 shows that if the probability δ that any Walsh mode fails to be integrated exactly is less than 1/8, then the probability that the median deviates by more than ε_K/δ decays as (8δ)^r, i.e., exponentially in the number of replicates r. Here ε_K aggregates the contributions of all Walsh modes outside a carefully chosen index set K. By selecting K based on the decay of Walsh coefficients— which encode both smoothness and effective dimensionality— the authors obtain error bounds that shrink at a rate O(n^{‑k‑½+η}) for any η > 0, where k denotes the order of mixed partial derivatives assumed to exist.

A central contribution is the introduction of an integrand‑specific, probabilistic error criterion rather than the worst‑case Sobolev‑ball norm traditionally used in tractability analysis. The paper defines effective dimensionality through the ANOVA decomposition of the integrand and a relative variation measure γ_u for each subset of variables u. When γ_u decays sufficiently fast (the “low effective dimensionality” condition), the constant C in the convergence bound becomes independent of the nominal dimension s, establishing strong tractability.

The theoretical development proceeds in several steps. Section 2 reviews digital net construction, random linear scrambling, and the complete random design (where all generator matrix entries are i.i.d. Bernoulli). Walsh functions are used to expand the integrand, leading to an exact error representation (Lemma 1) involving indicator variables Z(k) that signal failure of exact integration for a Walsh mode, and random signs S(k) from digital shifts. Section 3 formalizes the ANOVA decomposition and defines fractional Vitali variation to quantify the variation of each ANOVA component. Section 4 derives the median‑of‑means error bound under the simplified complete random design, establishing the exponential tail for the median error. Section 5 extends the analysis to the more realistic random linear scrambling, proving Theorem 5: for functions in the mixed‑smoothness class C(k,…,k) and under the low‑effective‑dimensionality condition on γ_u, the median estimator achieves dimension‑independent convergence of order O(n^{‑k‑½+η}) with a constant C that does not depend on s.

The authors compare their results with prior work on median digital nets (e.g.,