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 �Ônorms with � Ô �). ¯ extension of results on the covering numbers of convex hulls of basis functions to Ô-convex hulls ( � Ô �); ¯ an appendix containing the tightest known bounds on the entropy numbers of the identity operator between � Ò Ô and � Ò Ô ( � Ô � Ô �).