On the optimization of discrepancy measures

TL;DR

This paper introduces the average squared discrepancy, a new criterion for low-discrepancy point sets that is computationally efficient, smooth, and avoids known pathologies of existing measures, improving optimization outcomes.

Contribution

The paper proposes the average squared discrepancy, which is computationally feasible, equivalent to a weighted $L_2$ measure, and superior to traditional discrepancy measures in avoiding pathologies.

Findings

The average squared discrepancy can be computed in $O(dn^2)$ time.

It is equivalent to a weighted symmetric $L_2$ criterion.

Optimizing this measure yields strong performance for the $L_2$ star discrepancy.

Abstract

Points in the unit cube with low discrepancy can be constructed using algebra or, more recently, by direct computational optimization of a criterion. The usual $L_\infty$ star discrepancy is a poor criterion for this because it is computationally expensive and lacks differentiability. Its usual replacement, the $L_2$ star discrepancy, is smooth but exhibits other pathologies shown by J. Matou\v{s}ek. In an attempt to address these problems, we introduce the \textit{average squared discrepancy} which averages over $2^d$ versions of the $L_2$ star discrepancy anchored in the different vertices of $[0,1]^d$. Not only can this criterion be computed in $O(dn^2)$ time, like the $L_2$ star discrepancy, but also we show that it is equivalent to a weighted symmetric $L_2$ criterion of Hickernell's by a constant factor. We compare this criterion with a wide range of traditional discrepancy measures, and show that only the average squared discrepancy avoids the problems raised by Matou\v{s}ek. Furthermore, we present a comprehensive numerical study showing in particular that optimizing for the average squared discrepancy leads to strong performance for the $L_2$ star discrepancy, whereas the converse does not hold.