Sharp regret-Hellinger bounds for Gaussian empirical Bayes via polynomial approximation

Jiafeng Chen and Yihong Wu

Papers from arXiv.org

Abstract: A central problem in the theory of empirical Bayes is to control the regret (excess risk) of a learned Bayes rule by the Hellinger distance between the estimated and true marginal densities. In the normal means model, the classical result of Jiang and Zhang (2009, Annals of Statistics) achieves this only after regularizing the Bayes rule and incurs an extraneous cubic logarithmic factor through a delicate recursive argument. This paper introduces a new technique, based on polynomial approximation and Bernstein-type inequalities for weighted $L_2$ norms, that bounds the unregularized regret directly. The method is conceptually simpler and yields sharper, sometimes optimal, regret bounds. For compactly supported priors, we prove the sharp bound that the regret is at most $O(\epsilon^2 \log(1/\epsilon)/\log\log(1/\epsilon))$, where $\epsilon$ is the Hellinger distance between the marginal densities. The same method also extends to priors with exponential tails. Conversely, we show that regularization is genuinely necessary for heavy-tailed priors under only bounded moment assumptions. As a statistical consequence, we obtain improved regret bounds for the nonparametric maximum likelihood estimator.

Date: 2026-05
References: Add references at CitEc
Citations:

Downloads: (external link)
http://arxiv.org/pdf/2605.02070 Latest version (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:2605.02070

Access Statistics for this paper

More papers in Papers from arXiv.org
Bibliographic data for series maintained by arXiv administrators ().