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**Characterizations of learnability for classes of \(\{0,\dots,n\}\)-valued functions.**
*(English)*
Zbl 0827.68095

Summary: We investigate the PAC learnability of classes of \(\{0, \dots, n\}\)- valued functions \((n < \infty)\). For \(n = 1\) it is known that the finiteness of the Vapnik-Chervonenkis dimension is necessary and sufficient for learning. For \(n > 1\) several generalizations of the VC- dimension, each yielding distinct characterization of learnability, have been proposed by a number of researchers. 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 all these 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.

### MSC:

68T05 | Learning and adaptive systems in artificial intelligence |