We consider the problem of empirical Bayes estimation for (multivariate) Poisson means. Existing solutions that have been shown theoretically optimal for minimizing the regret (excess risk over the Bayesian oracle that knows the prior) have several shortcomings. For example, the classical Robbins estimator does not retain the monotonicity property of the Bayes estimator and performs poorly under moderate sample size. Estimators based on the minimum distance and non-parametric maximum likelihood (NPMLE) methods correct these issues, but are computationally expensive with complexity growing exponentially with dimension. Extending the approach of Barbehenn and Zhao (2022), in this work we construct monotone estimators based on empirical risk minimization (ERM) that retain similar theoretical guarantees and can be computed much more efficiently. Adapting the idea of offset Rademacher complexity Liang et al. (2015) to the non-standard loss and function class in empirical Bayes, we show that the shape-constrained ERM estimator attains the minimax regret within constant factors in one dimension and within logarithmic factors in multiple dimensions.

翻译：本文研究（多元）泊松均值估计中的经验贝叶斯问题。现有理论证明在最小化遗憾（相对于已知先验的贝叶斯神谕的超额风险）方面最优的解决方案存在若干缺陷。例如，经典的Robbins估计器未能保留贝叶斯估计器的单调性，且在中等样本量下表现不佳。基于最小距离和非参数最大似然（NPMLE）方法的估计器虽能修正这些问题，但其计算复杂度随维度呈指数增长。本研究沿用Barbehenn与Zhao（2022）的方法，构建了基于经验风险最小化（ERM）的单调估计器，该估计器在保持类似理论保证的同时，可实现更高效的计算。通过将Liang等人（2015）的偏移Rademacher复杂度思想适配至经验贝叶斯中的非标准损失函数与函数类，我们证明了形状约束的ERM估计器在一维情况下可在常数因子内达到极小极大遗憾，在多维情况下则在对数因子内达到该目标。