CONVEX REGRESSION IN MULTIDIMENSIONS: SUBOPTIMALITY OF LEAST SQUARES ESTIMATORS

• 成果类型: Article
• 署名作者: Kur, Gil; Gao, Fuchang; Guntuboyina, Adityanand; Sen, Bodhisattva
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich; University of Idaho; University of California System; University of California Berkeley; Columbia University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2445
• 发表日期: 2024
• 页码: 2791-2815
• 关键词: nonparametric approach; convergence-rates; risk bounds; Consistency; entropy; sets

摘要

Under the usual nonparametric regression model with Gaussian errors, are shown to be suboptimal for estimating a d-dimensional convex function in squared error loss when the dimension d is 5 or larger. The specific function classes considered include: (i) bounded convex functions supported on a polytope (in random design), (ii) Lipschitz convex functions supported on any convex domain (in random design) and (iii) convex functions supported on a polytope (in fixed design). For each of these classes, the risk of the LSE is proved to be of the order n-2/d (up to logarithmic factors) while the minimax risk is n-4/(d+4), when d >= 5. In addition, the first rate of convergence results (worst case and adaptive) for the unrestricted convex LSE are established in fixed design for polytopal domains for all d >= 1. Some new metric entropy results for convex functions are also proved, which are of independent interest.