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arXiv:2609.10921

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Ekene Ezeunala, Agastya Vibhuti Jha, Haotian Jiang

Published Sep 10, 2026• 0 citations
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Abstract

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(σ_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $σ_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

DOI
10.48550/arxiv.2609.10921

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Quasi-Monte Carlo Beyond Hardy-Krause II: (1+ε)n(1 + \varepsilon)n(1+ε)n Samples Suffice
Numerical integration studies how well one can estimate the integral of a function fff over [0,1)d[0,1)^d[0,1)d using nnn sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error O~d(σSO(f)/n)\widetilde{O}_d(σ_{\mathsf{SO}}(f)/n)Od​(σSO​(f)/n), where the smoothed-out variation σSO(f)σ_{\mathsf{SO}}(f)σSO​(f) can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires n2n^2n2 i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant ε>0\varepsilon > 0ε>0, we show that (1+ε)n(1+\varepsilon)n(1+ε)n i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.
We examine the problem of numerically estimating the integral of a function f. The classical approaches to this problem are Monte Carlo (MC) and quasi-Monte Carlo (QMC) methods. MC methods use random samples to evaluate f and have error , 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 Õ (σHK(f )/n ), where σΗΚ(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.
Numerical integration---approximating the integral of a function fff using nnn point evaluations---is a central task in science and engineering. The two main paradigms for this problem, the Monte Carlo and quasi-Monte Carlo methods, have distinct strengths and limitations, and a fundamental question is to design a method that combines the benefits of both. \smallskip Building on recent algorithmic advances in discrepancy theory, Bansal and Jiang \cite{BJ25a} gave a randomized QMC method that naturally bridges the MC and QMC error guarantees. Their method also achieves a surprising improvement over the classical Koksma--Hlawka inequality for QMC methods: it attains an error bound of O~(σSO(f)/n)\widetilde{O}(σ_{\mathsf{SO}}(f)/n)O(σSO​(f)/n), where σSO(f)σ_{\mathsf{SO}}(f)σSO​(f) is a new notion of \emph{smoothed-out variation} that they introduced and showed to be substantially smaller than the Hardy--Krause variation governing the classical bound. \smallskip However, the analysis in \cite{BJ25a} is quite involved: it must carefully exploit the structure of the dyadic decomposition and the randomness of the algorithm inside a sufficiently fine discretization of the Hlawka--Zaremba formula to obtain cancellations among the high-frequency components in the Fourier decomposition of fff. The contribution of this article is twofold: (1) We give an equivalent characterization of σSO(f)σ_{\mathsf{SO}}(f)σSO​(f) in terms of the Haar--Besov seminorm of fff, relating this new notion of smoothed-out variation to classical quantities. (2) Through this characterization, we provide a conceptually simpler and more direct analysis of the Bansal--Jiang QMC method via Haar decomposition, bypassing the use of the Hlawka--Zaremba formula, Fourier decomposition, and the delicate cancellation arguments of \cite{BJ25a} that heavily exploit the structure of dyadic decomposition.
Abstract A crude Monte Carlo (MC) method allows to calculate integrals over a d-dimensional cube. As the number N of integration nodes becomes large, the rate of probable error of the MC method decreases as O ⁢ ( 1 / N ) {O(1/\sqrt{N})} . The use of quasi-random points instead of random points in the MC algorithm converts it to the quasi-Monte Carlo (QMC) method. The asymptotic error estimate of QMC integration of d-dimensional functions contains a multiplier 1 / N {1/N} . However, the multiplier ( ln ⁡ N ) d {(\ln N)^{d}} is also a part of the error estimate, which makes it virtually useless. We have proved that, in the general case, the QMC error estimate is not limited to the factor 1 / N {1/N} . However, our numerical experiments show that using quasi-random points of Sobol sequences with N = 2 m {N=2^{m}} with natural m makes the integration error approximately proportional to 1 / N {1/N} . In our numerical experiments, d ≤ 15 {d\leq 15} , and we used N ≤ 2 40 {N\leq 2^{40}} points generated by the SOBOLSEQ16384 code published in 2011. In this code, d ≤ 2 14 {d\leq 2^{14}} and N ≤ 2 63 {N\leq 2^{63}} .
Monte Carlo (MQ) method is a powerful tool to approximate high dimensional integrals. The disadvantage of ordinary MQ is the slow convergence rate by cause of the essential randomness of this method. Computations of this convergence can lead to pure time consuming. Quasi-Monte Carlo (QMC) method yields considerably better results since this method is a deterministic alternative which uses Low-Discrepancy sequences instead of random samples . There are a multiplicity of big open problems in QMC-methods, problems partly arising from applications and partly arising from theory. QMC are developed to integrate over unite cube, where it has much more accuracy than MC for integrands of bounded variation. Integration over more general spaces such as triangles, disks and Cartesian products of such spaces is more challenging for QMC. Nevertheless in real-world applications various problems are defined over such spaces. The aim of this thesis is to provide a survey of a solution of such problems of numerical integration defined over non-cubical spaces. We present QMC and randomised QMC (RQMC) constructions in the triangle with a vanishing discrepancy based on the recent work of Basu 2016. The QMC construction is a version of the Van der Corput-Halton sequences specially made to the unit triangle. The attraction of scrambled net is replication based error estimation for QMC with slightly the same accuracy as QMC, and for smooth enough integrands.
Quasi-Monte Carlo (QMC) methods are known to achieve faster convergence rates than Monte Carlo (MC), but their effectiveness in high dimensions often relies on additional structure, such as low effective dimension or carefully chosen coordinate weights. Moreover, in many applications one has access only to random samples rather than deterministic QMC constructions. In this work, we extend the recent method of N. Bansal and H. Jiang and construct high-dimensional QMC point sets from random samples via combinatorial discrepancy. We establish error bounds for these constructions in weighted function spaces, including settings with low effective dimension in both the superposition and truncation senses. We also implement the resulting Subgaussian Transference algorithm and present numerical experiments to assess empirically the performance of these constructions.
The commonly quoted error rates for QMC integration with an infinite low discrepancy sequence is O(n−1log⁡(n)r)O(n^{-1}\log(n)^r)O(n−1log(n)r) with r=dr=dr=d for extensible sequences and r=d−1r=d-1r=d−1 otherwise. Such rates hold uniformly over all ddd dimensional integrands of Hardy-Krause variation one when using nnn evaluation points. Implicit in those bounds is that for any sequence of QMC points, the integrand can be chosen to depend on nnn. In this paper we show that rates with any r1r1r1 is needed. An example with d=3d=3d=3 and nnn up to 21002^{100}2100 might possibly require r>1r>1r>1.