A Simpler Analysis of the Bansal-Jiang Quasi Monte-Carlo Algorithm via Haar Wavelets
Data Structures and Algorithms
Summary
The authors explain and simplify a recent advanced method by Bansal and Jiang for approximating integrals using a mix of Monte Carlo and quasi-Monte Carlo techniques. Bansal and Jiang introduced a new way to measure how smooth a function is, called smoothed-out variation, which leads to better error estimates than classical methods. This paper shows that this new measure can be understood using a known mathematical tool called the Haar--Besov seminorm. By doing this, the authors provide a clearer and more straightforward proof of the original results, avoiding complex formulas and tricky arguments used in the prior work.
Authors
Jiaheng Cheng, Agastya Vibhuti Jha, Haotian Jiang
Abstract
Numerical integration---approximating the integral of a function $f$ using $n$ 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 $\widetilde{O}(σ_{\mathsf{SO}}(f)/n)$, where $σ_{\mathsf{SO}}(f)$ 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 $f$. The contribution of this article is twofold: (1) We give an equivalent characterization of $σ_{\mathsf{SO}}(f)$ in terms of the Haar--Besov seminorm of $f$, 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.