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.

翻译：L∞星偏差是衡量点集在给定空间中均匀性的指标。低星偏差点集被用作实验设计、贝叶斯优化算法的初始设计、拟蒙特卡洛积分方法以及许多其他应用。近期研究表明，当考虑固定规模的点集而非其收敛行为时，Sobol'、Halton或Hammersley序列等经典构造方法可能被大幅超越。这些与前述应用高度相关的结果引发了一个问题：现有构造方法通过规模特定优化能提升多少？本文针对所谓的克罗内克构造法研究这一问题。聚焦三维场景，我们表明优化该构造的两个可配置参数可产生至少优于500个点集的当前最优值。利用算法配置技术irace，我们进一步推导出参数，从而为整个点集规模区间提供新的当前最优偏差值。