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 ¼ Ô ½). ¯ extension of results on the covering numbers of convex hulls of basis functions to Ô-convex hulls (¼ Ô ½); ¯ an appendix containing the tightest known bounds on the...

• Type: Conference
• Year: 2000
• Where: COLT
• Authors: Robert C. Williamson, Alex J. Smola, Bernhard Schölkopf
• Keywords: COLT 2000, Generalization Performance, Machine Learning, Maximum Margin Classes, Tightest Known Bounds