Entropy Numbers of Linear Function Classes

Abstract

This paper collects together a miscellany of results originally motivated by the analysis of the generalization performance of the "maximum-margin" algorithm due to Vapnik and others. The key feature of the paper is its operator-theoretic viewpoint. New bounds on covering numbers for classes related to Maximum Margin classes are derived directly without making use of a combinatorial dimension such as the VC-dimension. Specific contents of the paper include: a new and self-contained proof of Maurey's theorem and some generalizations with small explicit values of constants; bounds on the covering numbers of maximum margin classes suitable for the analysis of their generalization performance; the extension of such classes to those induced by balls in quasi-Banach spaces (such as ` p - norms with 0 < p < 1). extension of results on the covering numbers of convex hulls of basis functions to p-convex hulls (0 < p 1); an appendix containing the tightest known bounds on the e...