Characterizations of Learnability for Classes of {0, . . . , n}-valued Functions

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 and for any "reasonable" loss function