Scale-sensitive Dimensions, Uniform Convergence, and Learnability

Learnability in Valiant’s PAC learning model has been shown to be strongly related to the existence of uniform laws of large numbers. These laws define a distribution-free convergence property of means to expectations uniformly over classes of random variables. Classes of real-valued functions enjoying such a property are also known as uniform Glivenko–Cantelli classes. In this paper, we prove, through a generalization of Sauer’s lemma that may be interesting in its own right, a new characterization of uniform Glivenko–Cantelli classes. Our characterization yields Dudley, Gine´, and Zinn’s previous characterization as a corollary. Furthermore, it is the first based on a simple combinatorial quantity generalizing the Vapnik–Chervonenkis dimension. We apply this result An earlier version of this paper appeared in Proceedings of the 34th Annual Symposium on the

• Type: Conference
• Year: 1993
• Where: FOCS
• Authors: Noga Alon, Shai Ben-David, Nicolò Cesa-Bianchi, David Haussler
• Keywords: Distribution-free Convergence Property, FOCS 1993, Theoretical Computer Science, Uniform Glivenko–cantelli Classes, Uniform Laws