Sharp regret-Hellinger bounds for Gaussian empirical Bayes via polynomial approximation (opens in new tab)

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 introdu...