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

Imène Ait Abderrahim, Carola Doerr, Martin Durand

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.

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

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.

Paper Structure

This paper contains 14 sections, 4 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions and outline of the paper
  4. Preliminaries
  5. $L_{\infty}$ star discrepancy
  6. Kronecker point sets
  7. Optimizing Kronecker point set generation
  8. Experimental results
  9. Assessing the effectiveness of CMA-ES and parameter tuning for low discrepancy
  10. Using post-processing on Kronecker point sets.
  11. Assesing Parameter tuning on different $n$ intervals
  12. Assessing the Kronecker method for dimension $D = 4$
  13. Discussion
  14. Conclusion

Figures (4)

  • Figure 1: Heatmaps. The $X$ (resp. $Y$) coordinate is the $p_2\xspace$ (resp. $p_3\xspace$) value, the color indicates the $L_{\infty}$ star discrepancy of the 3D point set generated with the $(1/n\xspace,p_2\xspace,p_3\xspace)$ parameters, for $n\xspace=100$. The second and last heatmaps contain only points for which the Kronecker point set has a discrepancy below 0.055 and 0.045, respectively.
  • Figure 2: Plot of the $L_{\infty}$ discrepancy for $n$ points
  • Figure 3: Plot of the inverse $L_{\infty}$ star discrepancy for $n$ between 5 and 2500
  • Figure 4: $L_{\infty}\xspace$ discrepancy with Parameters tuning for $n$ small intervals