Quasi-Monte Carlo Beyond Hardy-Krause

TL;DR

The paper addresses the fundamental trade-off between Monte Carlo and quasi-Monte Carlo methods for numerical integration. It introduces a randomized transference-based algorithm, SubgTransference, that builds low-discrepancy point sets with subgaussian properties and achieves an error bound of order $\tilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where $\sigma_{\mathsf{SO}}(f)$ is a new smoothed variation that sits between the standard deviation and Hardy–Krause variation. This approach combines the best aspects of MC and QMC—flexibility, efficiency, and statistical error control—while surpassing traditional KH-type guarantees, and it does so with amortized $\tilde{O}_d(1)$ time per sample. The authors also provide a detailed 1-D and higher-dimensional analysis, define $\sigma_{\mathsf{SO}}$ in general terms, and discuss open problems for reducing sample requirements further.

Abstract

The classical approaches to numerically integrating a function $f$ are Monte Carlo (MC) and quasi-Monte Carlo (QMC) methods. MC methods use random samples to evaluate $f$ and have error $O(σ(f)/\sqrt{n})$, where $σ(f)$ is the standard deviation of $f$. QMC methods are based on evaluating $f$ at explicit point sets with low discrepancy, and as given by the classical Koksma-Hlawka inequality, they have error $\widetilde{O}(σ_{\mathsf{HK}}(f)/n)$, where $σ_{\mathsf{HK}}(f)$ is the variation of $f$ in the sense of Hardy and Krause. These two methods have distinctive advantages and shortcomings, and a fundamental question is to find a method that combines the advantages of both. In this work, we give a simple randomized algorithm that produces QMC point sets with the following desirable features: (1) It achieves substantially better error than given by the classical Koksma-Hlawka inequality. In particular, it has error $\widetilde{O}(σ_{\mathsf{SO}}(f)/n)$, where $σ_{\mathsf{SO}}(f)$ is a new measure of variation that we introduce, which is substantially smaller than the Hardy-Krause variation. (2) The algorithm only requires random samples from the underlying distribution, which makes it as flexible as MC. (3) It automatically achieves the best of both MC and QMC (and the above improvement over Hardy-Krause variation) in an optimal way. (4) The algorithm is extremely efficient, with an amortized $\widetilde{O}(1)$ runtime per sample. Our method is based on the classical transference principle in geometric discrepancy, combined with recent algorithmic innovations in combinatorial discrepancy that besides producing low-discrepancy colorings, also guarantee certain subgaussian properties. This allows us to bypass several limitations of previous works in bridging the gap between MC and QMC methods and go beyond the Hardy-Krause variation.

Figures (1)

• Figure 1: Placements of 64 points in $[0,1]^2$ (from Mat09). On the left are random points. In the middle, the grid points have large discrepancy as the green rectangle has area $1/\sqrt{n}$ but no points. On the right is the Van der Corput set, which already visually looks more "uniform".

Theorems & Definitions (38)

• Theorem 1.1
• Remark 1.2: Smoothness Assumptions
• Remark 1.3: General Form of $\sigma_{\mathsf{SO}}$
• Theorem 1.4: Best of MC and Hardy-Krause
• Theorem 1.5: Best of MC and $\sigma_{\SO}$
• Lemma 2.1: Hlawka–Zaremba Formula, Hla61Zar68
• Theorem 2.2: $\ell_2$-Koksma-Hlawka Inequality, Kok42Hla61Zar68
• Definition 2.3: Subgaussian Vectors
• Theorem 2.4: ALS21
• Lemma 2.4: Properties of Structured Decomposition Matrix