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

Zexin Pan

TL;DR

The paper tackles the challenge of achieving dimension-independent convergence in high-dimensional QMC by applying a median-of-means strategy to linearly scrambled base-2 digital nets. It combines Walsh-function analysis with ANOVA-based effective dimensionality to derive probabilistic error guarantees that yield near-optimal, integrand-specific convergence rates under weak assumptions. The authors establish strong tractability under low effective dimensionality and extend the results to general randomization schemes, including both completely random designs and random linear scrambling, supported by numerical experiments. The findings offer a robust, parameter-light approach to high-dimensional integration with practical implications for applications in finance and physics, where dimension often grows without bound. Overall, the work provides a rigorous foundation for dimension-generalizable convergence using median-based RQMC, bridging probabilistic guarantees with tractable, real-world performance.

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.

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

TL;DR

The paper tackles the challenge of achieving dimension-independent convergence in high-dimensional QMC by applying a median-of-means strategy to linearly scrambled base-2 digital nets. It combines Walsh-function analysis with ANOVA-based effective dimensionality to derive probabilistic error guarantees that yield near-optimal, integrand-specific convergence rates under weak assumptions. The authors establish strong tractability under low effective dimensionality and extend the results to general randomization schemes, including both completely random designs and random linear scrambling, supported by numerical experiments. The findings offer a robust, parameter-light approach to high-dimensional integration with practical implications for applications in finance and physics, where dimension often grows without bound. Overall, the work provides a rigorous foundation for dimension-generalizable convergence using median-based RQMC, bridging probabilistic guarantees with tractable, real-world performance.

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.

Paper Structure

This paper contains 11 sections, 14 theorems, 152 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Digital net construction and randomization
  4. Walsh functions and error analysis for base-2 digital nets
  5. Probabilistic error guarantee of the median estimator
  6. Walsh coefficients and function smoothness
  7. ANOVA decomposition and effective dimensionality
  8. Tractability under the completely random design
  9. Tractability under more general randomization
  10. Numerical experiments
  11. Discussion

Key Result

Theorem 1

Given $\eta>0$, $k\in \mathbb{N}$ and $f\in C^{(k,\dots,k)}([0,1]^s)$, it holds with probability $1-2^{-r}$ that the median estimator $\hat{\mu}^{(r)}_{\infty}$ from Subsection subsec:mse satisfies The constant $C_{\eta,k,f}$ is independent of $s$ under the random linear scrambling if the relative variation $\gamma_u$ satisfies the tractability conditions in Theorem thm:tractability.

Figures (2)

  • Figure 1: Integration errors for $f_{s,\gamma,\alpha}$ with $\alpha=0$.
  • Figure 2: Integration errors for $f_{s,\gamma,\alpha}$ with $\alpha=1$.

Theorems & Definitions (35)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark 1
  • Lemma 5
  • ...and 25 more