On the optimization of discrepancy measures

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.