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

TL;DR

This paper improves the construction of 3D Kronecker point sets to achieve lower star discrepancy by optimizing parameters with algorithm configuration, outperforming existing methods for various set sizes.

Contribution

It demonstrates that size-specific parameter optimization of the 3D Kronecker construction can produce state-of-the-art low-discrepancy point sets.

Findings

Optimized parameters outperform previous discrepancy values for sets of at least 500 points.

Using irace, the authors find parameters that improve discrepancy across a range of sizes.

The approach sets new benchmarks for low star discrepancy in 3D point sets.

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.