On Learnability with Computable Learners

Abstract

We initiate a study of learning with computable learners and computable output predictors. Recent results in statistical learning theory have shown that there are basic learning problems whose learnability can not be determined within ZFC. This motivates us to consider learnability by algorithms with computable output predictors (both learners and predictors are then representable as finite objects). We thus propose the notion of CPAC learnability, by adding some basic computability requirements into a PAC learning framework. As a first step towards a characterization, we show that in this framework learnability of a binary hypothesis class is not implied by finiteness of its VC-dimension anymore. We also present some situations where we are guaranteed to have a computable learner.

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