Computing Star Discrepancies with Numerical Black-Box Optimization Algorithms

TL;DR

This paper evaluates the effectiveness of various black-box optimization algorithms in computing the $L_{ infty}$ star discrepancy of point sets, revealing their limitations and suggesting directions for future improvements.

Contribution

It systematically compares 8 black-box optimization algorithms on discrepancy computation, highlighting their poor performance and providing a parallel implementation of a known algorithm.

Findings

All optimizers perform poorly on most instances.

Random search often outperforms sophisticated algorithms.

Current methods struggle to capture the problem's global structure.

Abstract

The $L_{\infty}$ star discrepancy is a measure for the regularity of a finite set of points taken from $[0,1)^d$. Low discrepancy point sets are highly relevant for Quasi-Monte Carlo methods in numerical integration and several other applications. Unfortunately, computing the $L_{\infty}$ star discrepancy of a given point set is known to be a hard problem, with the best exact algorithms falling short for even moderate dimensions around 8. However, despite the difficulty of finding the global maximum that defines the $L_{\infty}$ star discrepancy of the set, local evaluations at selected points are inexpensive. This makes the problem tractable by black-box optimization approaches. In this work we compare 8 popular numerical black-box optimization algorithms on the $L_{\infty}$ star discrepancy computation problem, using a wide set of instances in dimensions 2 to 15. We show that all used optimizers perform very badly on a large majority of the instances and that in many cases random search outperforms even the more sophisticated solvers. We suspect that state-of-the-art numerical black-box optimization techniques fail to capture the global structure of the problem, an important shortcoming that may guide their future development. We also provide a parallel implementation of the best-known algorithm to compute the discrepancy.