Improved Distribution Estimation in $\ell_\infty$

2026-05-28

AI Summary

This work addresses the minimax estimation of discrete probability distributions under the $\ell_\infty$ norm, aiming to characterize optimal risk bounds both in expectation and with high probability. By integrating minimax theory, high-dimensional probability analysis, and empirical process techniques with constructive proofs, the study establishes the first fully computable, data-dependent tight risk bound, thereby resolving an open problem posed by Kontorovich and Painsky. The analysis also precisely identifies the structure of extremal distributions that achieve worst-case risk. These theoretical advances not only sharpen existing $\ell_\infty$ risk bounds but also lead to estimators that demonstrate superior empirical performance compared to current methods.

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.

Problem

• distribution estimation
• ℓ∞ norm
• minimax bounds
• extremal distribution
• high-probability tail bounds

Innovation

• distribution estimation
• ℓ∞ norm
• minimax bounds
• extremal distribution
• high-probability tail bounds