Valiant's protocol for learning is extended to the case where the distribution of the examples is known to the learner. Namely, the notion of a concept class C being learnable with respect to distribution D is defined and the learnable pairs (C, D) of concept classes C and distributions D are characterized. Another notion is the existence of a finite cover for C with respect to D. The main result is that C is learnable with respect to D if and only if C is finitely coverable with respect to D. The size of the cover is then related to the Vapnik-Chervonenkis dimension. An additional property of the learning method is robustness, i.e., learning succeeds even if part of the input is erroneous. It is also shown that if D is discrete then every concept class is learnable with respect to D. The main concern of the paper is the number of examples sufficient to probabilistically identify (or approximate) a concept-not the time needed to compute it. Indeed, in some cases the function which associates a sample with a hypothesis is undecidable, and even if it is computable, the computation may be infeasible. The computational complexity of the algorithms used for learning are considered only for discrete distributions.
REF show that if the distribution is known to the learner, a specific hypothesis class is learnable if and only if there is a finite ǫ-cover of this hypothesis class with respect to the distribution.
Learnability with respect to fixed distributions
Theor. Comput. Sci.
Theor. Comput. Sci.
Computer Science