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 $O(ε^2 \log(1/ε)/\log\log(1/ε))$, where $ε$ 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.

翻译：经验贝叶斯理论中的一个核心问题是通过估计边际密度与真实边际密度之间的海林格距离来控制学习到的贝叶斯规则的遗憾（超额风险）。在正态均值模型中，Jiang 和 Zhang（2009，《统计年鉴》）的经典结果仅在对贝叶斯规则进行正则化后实现了这一点，并通过一个精细的递归论证引入了额外的三次对数因子。本文提出了一种新技术，基于多项式逼近和加权 $L_2$ 范数的伯恩斯坦型不等式，直接界定了未正则化的遗憾。该方法在概念上更简单，并给出了更锐利、有时最优的遗憾界。对于紧支撑先验，我们证明了遗憾的锐界为 $O(ε^2 \log(1/ε)/\log\log(1/ε))$，其中 $ε$ 是边际密度之间的海林格距离。同一方法也延伸至指数尾先验。相反，我们表明在仅有有界矩假设下，对于重尾先验，正则化是真正必要的。作为统计推论，我们得到了非参数极大似然估计的改进遗憾界。