标题： 关于（正则化）经验风险最小化的集中性 显示英文标题

标题： On concentration for (regularized) empirical risk minimization

摘要： 经验风险最小化器的收敛速度在文献中已有广泛研究。 本文旨在提供一组互补的结果，特别是通过表明在归一化后，经验最小化器的风险集中在单个点上。 对于正态序列模型中的约束估计量，~\cite{chatterjee2014new}已经建立了此类结果。 我们首先将这一结果推广并细化到具有凸惩罚的正则最小二乘法，利用基于 Borell 定理的“直接”论证方法。 然后，我们研究其他损失函数的推广情况，包括指数族的负对数似然结合严格凸的正则化惩罚。 在此一般情况下，结果基于更“间接”的论证以及经验过程极大值的集中不等式。 显示英文摘要

摘要： Rates of convergence for empirical risk minimizers have been well studied in the literature. In this paper, we aim to provide a complementary set of results, in particular by showing that after normalization, the risk of the empirical minimizer concentrates on a single point. Such results have been established by~\cite{chatterjee2014new} for constrained estimators in the normal sequence model. We first generalize and sharpen this result to regularized least squares with convex penalties, making use of a "direct" argument based on Borell's theorem. We then study generalizations to other loss functions, including the negative log-likelihood for exponential families combined with a strictly convex regularization penalty. The results in this general setting are based on more "indirect" arguments as well as on concentration inequalities for maxima of empirical processes.