OPTIMAL BOUNDS FOR AGGREGATION OF AFFINE ESTIMATORS

• 成果类型: Article
• 署名作者: Bellec, Pierre C.
• 署名单位: Institut Polytechnique de Paris; ENSAE Paris; Rutgers University System; Rutgers University New Brunswick; Rutgers University System; Rutgers University New Brunswick
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1540
• 发表日期: 2018
• 页码: 30-59
• 关键词: independent random-variables; Nonparametric Regression; least-squares; variance-estimation; tail probabilities; convex aggregation; density estimators; Sparse Estimation; quadratic-forms; model selection

摘要

We study the problem of aggregation of estimators when the estimators are not independent of the data used for aggregation and no sample splitting is allowed. If the estimators are deterministic vectors, it is well known that the minimax rate of aggregation is of order log(M), where M is the number of estimators to aggregate. It is proved that for affine estimators, the minimax rate of aggregation is unchanged: it is possible to handle the linear dependence between the affine estimators and the data used for aggregation at no extra cost. The minimax rate is not impacted either by the variance of the affine estimators, or any other measure of their statistical complexity. The minimax rate is attained with a penalized procedure over the convex hull of the estimators, for a penalty that is inspired from the Q-aggregation procedure. The results follow from the interplay between the penalty, strong convexity and concentration.