Entropy Numbers of Linear Function Classes

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 `pnorms 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 entropy numbers of the identity operator between `np1 and `np2 (0 < p1 < p2 1).