Improved Distribution Estimation in $\ell_∞$

Doron Cohen, Aryeh Kontorovich, Yonatan Livshitz
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:21102-21115, 2026.

Abstract

We present improved bounds for estimating discrete probability distributions under the $\ell_{\infty}$ norm. These include minimax bounds in expectation and high-probability tail bounds. We resolve some of the open questions posed in Kontorovich and Painsky (JMLR, 2025) — including a fully empirical version of the tightest risk bound they presented and identifying the form of the worst-case extremal distribution. Encouraging empirical results are reported as well.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-cohen26c, title = {Improved Distribution Estimation in $\ell_∞$}, author = {Cohen, Doron and Kontorovich, Aryeh and Livshitz, Yonatan}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {21102--21115}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/cohen26c/cohen26c.pdf}, url = {https://proceedings.mlr.press/v306/cohen26c.html}, abstract = {We present improved bounds for estimating discrete probability distributions under the $\ell_{\infty}$ norm. These include minimax bounds in expectation and high-probability tail bounds. We resolve some of the open questions posed in Kontorovich and Painsky (JMLR, 2025) — including a fully empirical version of the tightest risk bound they presented and identifying the form of the worst-case extremal distribution. Encouraging empirical results are reported as well.} }
Endnote
%0 Conference Paper %T Improved Distribution Estimation in $\ell_∞$ %A Doron Cohen %A Aryeh Kontorovich %A Yonatan Livshitz %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-cohen26c %I PMLR %P 21102--21115 %U https://proceedings.mlr.press/v306/cohen26c.html %V 306 %X We present improved bounds for estimating discrete probability distributions under the $\ell_{\infty}$ norm. These include minimax bounds in expectation and high-probability tail bounds. We resolve some of the open questions posed in Kontorovich and Painsky (JMLR, 2025) — including a fully empirical version of the tightest risk bound they presented and identifying the form of the worst-case extremal distribution. Encouraging empirical results are reported as well.
APA
Cohen, D., Kontorovich, A. & Livshitz, Y.. (2026). Improved Distribution Estimation in $\ell_∞$. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:21102-21115 Available from https://proceedings.mlr.press/v306/cohen26c.html.

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