Median of Means Sampling for the Keister Function

TL;DR

This paper compares median-of-means and mean-of-means sampling methods for integrating the Keister function using RQMC, finding median-of-means more effective at larger sample sizes and confirming theoretical advantages.

Contribution

It provides empirical evidence that median-of-means sampling outperforms mean-of-means in high-dimensional RQMC integration for larger sample sizes.

Findings

Median-of-means outperforms mean-of-means for sample sizes >10^3

Mean-of-means is more accurate with smaller samples, especially for digital nets

Results confirm theoretical predictions about median-of-means' advantages

Abstract

This study investigates the performance of median-of-means sampling compared to traditional mean-of-means sampling for computing the Keister function integral using Randomized Quasi-Monte Carlo (RQMC) methods. The research tests both lattice points and digital nets as point distributions across dimensions 2, 3, 5, and 8, with sample sizes ranging from 2^8 to 2^19 points. Results demonstrate that median-of-means sampling consistently outperforms mean-of-means for sample sizes larger than 10^3 points, while mean-of-means shows better accuracy with smaller sample sizes, particularly for digital nets. The study also confirms previous theoretical predictions about median-of-means' superior performance with larger sample sizes and reflects the known challenges of maintaining accuracy in higher-dimensional integration. These findings support recent research suggesting median-of-means as a promising alternative to traditional sampling methods in numerical integration, though limitations in sample size and dimensionality warrant further investigation with different test functions and larger parameter spaces.