Problem
Research questions and friction points this paper is trying to address.
numerical integration
quasi-Monte Carlo
Hardy-Krause variation
Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice
🤖 AI Summary
本文解决了数值积分中样本数量过多的问题，通过改进的在线Haar-thinning方法，仅使用(1+ε)n个独立同分布样本即可实现超越Hardy-Krause界限的误差估计。
📝 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}.
Innovation
Methods, ideas, or system contributions that make the work stand out.
beyond-Hardy--Krause
low-discrepancy point sequences
(1+ε)n samples
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