# Improved Distribution Estimation in ℓ_∞

arXiv: https://arxiv.org/abs/2605.30509 · HTML: https://mlsanity.com/paper/2605.30509/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 8 references that are in the map.
- First posted on arXiv 28 May 2026.

## Builds on (most cited first, 3 of 3)

- [On the Non-asymptotic and Sharp Lower Tail Bounds of Random Variables](https://mlsanity.com/paper/1810.09006/) (2018)
- [Distribution Estimation under the Infinity Norm](https://mlsanity.com/paper/2402.08422/) (2024)
- [The Empirical Mean is Minimax Optimal for Local Glivenko-Cantelli](https://mlsanity.com/paper/2410.02835/) (2024)

## Abstract (arXiv)

We present improved bounds for estimating discrete probability distributions under the ℓ_∞ 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.