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

Zexin Pan

TL;DR

This work introduces a median‑of‑means approach for the base‑2 scrambled digital‑net QMC estimator, showing that the sample median of independent QMC runs converges at almost the optimal rate across several natural smoothness classes without requiring prior knowledge of the function space or a predesigned net. By leveraging the Walsh decomposition and a carefully defined event controlling outliers, the authors prove almost‑optimal RMSE decay under bounded variation, fractional Vitali variation, and fractional Sobolev space assumptions, with rates improving by up to a half‑power relative to deterministic bounds. The numerical experiments corroborate the theory, illustrating robust, automatic convergence in both low and high dimensions and across varying smoothness, while highlighting practical considerations between completely random designs and scrambling. Overall, the median trick provides a practical, flexible, and broadly applicable mechanism to achieve fast convergence in randomized QMC without delicate parameter tuning or specialized net design, with implications for uncertainty quantification and high‑dimensional integration.

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.

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

TL;DR

This work introduces a median‑of‑means approach for the base‑2 scrambled digital‑net QMC estimator, showing that the sample median of independent QMC runs converges at almost the optimal rate across several natural smoothness classes without requiring prior knowledge of the function space or a predesigned net. By leveraging the Walsh decomposition and a carefully defined event controlling outliers, the authors prove almost‑optimal RMSE decay under bounded variation, fractional Vitali variation, and fractional Sobolev space assumptions, with rates improving by up to a half‑power relative to deterministic bounds. The numerical experiments corroborate the theory, illustrating robust, automatic convergence in both low and high dimensions and across varying smoothness, while highlighting practical considerations between completely random designs and scrambling. Overall, the median trick provides a practical, flexible, and broadly applicable mechanism to achieve fast convergence in randomized QMC without delicate parameter tuning or specialized net design, with implications for uncertainty quantification and high‑dimensional integration.

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 -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.

Paper Structure

This paper contains 14 sections, 19 theorems, 135 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Digital nets and randomization
  4. Elementary intervals and the $t$ parameter
  5. Fourier-Walsh decomposition
  6. Mean squared error of the median
  7. Convergence rate of the median
  8. When $f$ has bounded variation of order $\alpha$
  9. When dominating mixed derivatives of $f$ up to order $\alpha$ are continuous with bounded fractional variation
  10. When $f$ is in the Sobolev space of order $\alpha+\lambda$
  11. Numerical experiment
  12. One-dimensional $f$ with discontinuous derivatives
  13. 20-dimensional $f$ with decaying variable importance
  14. Discussion

Key Result

Theorem 1

Let $f\in C([0,1]^s)$ and let $\boldsymbol{x}_i$ be defined by equation eqn:xequalMCiplusD for $i\in\mathbb{Z}_{<2^m}$. Then where

Figures (2)

  • Figure 1: Comparison of medians with two types of randomization and scrambled digital nets of order $1,2$ and $3$. The test functions are $f_{\alpha^*}$ for $\alpha^*=1$ (top), $\alpha^*=2$ (middle) and $\alpha^*=3$ (bottom). The theory predicts the approximate convergence rate is $O(n^{-\alpha-1})$ for medians with complete random designs ($\diamondsuit$) and medians with random linear scrambling ($\triangledown$), $O(n^{-1.5})$ for digital nets of order $1$ ($\times$), $O(n^{-\min(2.5,\alpha^*+1)})$ for digital nets of order 2 ($\triangle$) and $O(n^{-\min(3.5,\alpha^*+1)})$ for digital nets of order 3 ($\Circle$).
  • Figure 2: Comparison of medians with two types of randomization and scrambled digital nets of order $1,2$ and $3$. The test functions are $f_{c}$ for $c=0.5$ (top), $c=1.5$ (middle) and $c=2.5$ (bottom). The theory predicts the approximate convergence rate is $O(n^{-\lceil c\rceil-0.5})$ for medians with complete random designs ($\diamondsuit$) and medians with random linear scrambling ($\triangledown$), $O(n^{-1.5})$ for digital nets of order $1$ ($\times$), $O(n^{-\min(2.5,\lceil c\rceil+0.5)})$ for digital nets of order 2 ($\triangle$) and $O(n^{-\min(3.5,\lceil c\rceil+0.5)})$ for digital nets of order 3 ($\Circle$).

Theorems & Definitions (41)

  • Theorem 1
  • proof
  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 31 more