GENERAL NONEXACT ORACLE INEQUALITIES FOR CLASSES WITH A SUBEXPONENTIAL ENVELOPE

• 成果类型: Article
• 署名作者: Lecue, Guillaume; Mendelson, Shahar
• 署名单位: Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Universite Gustave-Eiffel; Centre National de la Recherche Scientifique (CNRS); Technion Israel Institute of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS965
• 发表日期: 2012
• 页码: 832-860
• 关键词: Empirical Processes; Lower bounds; risk bounds; Lasso; selection; sparsity; Operators; RECOVERY

摘要

We show that empirical risk minimization procedures and regularized empirical risk minimization procedures satisfy nonexact oracle inequalities in an unbounded framework, under the assumption that the class has a subexponential envelope function. The main novelty, in addition to the boundedness assumption free setup, is that those inequalities can yield fast rates even in situations in which exact oracle inequalities only hold with slower rates. We apply these results to show that procedures based on l(1) and nuclear norms regularization functions satisfy oracle inequalities with a residual term that decreases like 1/n forevery L-q-loss functions (q >= 2), while only assuming that the tail behavior of the input and output variables are well behaved. In particular, no RIP type of assumption or incoherence condition are needed to obtain fast residual terms in those setups. We also apply these results to the problems of convex aggregation and model selection.