PAC Learning, VC Dimension, and the Arithmetic Hierarchy

TL;DR

This paper characterizes the complexity of identifying PAC-learnable concept classes, showing they are m-complete Σ^0_3 within a broad class, and establishes equivalence between PAC learnability and finite VC dimension.

Contribution

It computes the exact arithmetical complexity of the index set of PAC-learnable classes and links PAC learnability with finite VC dimension within a computability framework.

Findings

PAC learnability index set is m-complete Σ^0_3

PAC learnability is equivalent to finite VC dimension for the considered classes

The class of concept classes covers all standard examples

Abstract

We compute that the index set of PAC-learnable concept classes is $m$-complete $\Sigma^0_3$ within the set of indices for all concept classes of a reasonable form. All concept classes considered are computable enumerations of computable $\Pi^0_1$ classes, in a sense made precise here. This family of concept classes is sufficient to cover all standard examples, and also has the property that PAC learnability is equivalent to finite VC dimension.