Characterizations of learnability for classes of { O, …, n }-valued functions
Shai Ben-David, Nicolò Cesa‐Bianchi, Philip M. Long · 1992
We investigate the PAC learnability of classes of {0,…,n}-valued functions. For n = 1, it is known that the finiteness of the Vapnik-Chervonenkis dimension is necessary and sufficient for learning. In this paper we present a general scheme for extending the VC-dimension to the case n > 1. Our scheme defines a wide variety of notions of dimension in which several variants of the VC-dimension, previously introduced in the context of learning, appear as special cases. Our main result is a simple condition characterizing the set of notions of dimension whose finiteness is necessary and sufficient for learning. This provides a variety of new tools for determining the learnability of a class of multi-valued functions. Our characterization is also shown to hold in the “robust” variant of PAC model.