{"ID":2886031,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.04926","arxiv_id":"2508.04926","title":"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. Matoušek. 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 Matoušek. 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.","short_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 $L_2$ s...","url_abs":"https://arxiv.org/abs/2508.04926","url_pdf":"https://arxiv.org/pdf/2508.04926v1","authors":"[\"François Clément\",\"Nathan Kirk\",\"Art B. Owen\",\"T. Konstantin Rusch\"]","published":"2025-08-06T23:20:48Z","proceeding":"math.NA","tasks":"[\"math.NA\",\"math.OC\"]","methods":"[]","has_code":false}