Finding Low Star Discrepancy 3D Kronecker Point Sets Using Algorithm Configuration Techniques

2026-04-01

AI Summary

This work addresses the high $L_\infty$ star discrepancy of three-dimensional Kronecker point sets at fixed sizes by proposing a parameter optimization approach based on the algorithm configuration framework irace. By automatically tuning the two generating parameters of Kronecker sequences, the method systematically searches for optimal configurations across various point set sizes, particularly those with at least 500 points. This study marks the first application of irace to the construction of Kronecker point sets and achieves significantly better performance than existing methods across multiple continuous size intervals, establishing new state-of-the-art records for $L_\infty$ star discrepancy. The resulting point sets exhibit enhanced uniformity, offering improved suitability for applications such as experimental design, Bayesian optimization, and quasi-Monte Carlo integration.

Abstract

The L infinity star discrepancy is a measure for how uniformly a point set is distributed in a given space. Point sets of low star discrepancy are used as designs of experiments, as initial designs for Bayesian optimization algorithms, for quasi-Monte Carlo integration methods, and many other applications. Recent work has shown that classical constructions such as Sobol', Halton, or Hammersley sequences can be outperformed by large margins when considering point sets of fixed sizes rather than their convergence behavior. These results, highly relevant to the aforementioned applications, raise the question of how much existing constructions can be improved through size-specific optimization. In this work, we study this question for the so-called Kronecker construction. Focusing on the 3-dimensional setting, we show that optimizing the two configurable parameters of its construction yields point sets outperforming the state-of-the-art value for sets of at least 500 points. Using the algorithm configuration technique irace, we then derive parameters that yield new state-of-the-art discrepancy values for whole ranges of set sizes.

Problem

• star discrepancy
• Kronecker point sets
• algorithm configuration
• quasi-Monte Carlo
• design of experiments

Innovation

• star discrepancy
• Kronecker point sets
• algorithm configuration
• irace
• quasi-Monte Carlo