PAC-learning is Undecidable

TL;DR

This paper proves that determining whether a concept is PAC-learnable is fundamentally undecidable, revealing a key theoretical limitation in the field of machine learning.

Contribution

It establishes that there is no algorithmic criterion to decide PAC-learnability, highlighting an inherent undecidability in the theoretical foundations of machine learning.

Findings

Proves PAC-learnability testing is Turing-undecidable.

Highlights fundamental limits in algorithmic learnability.

Discusses implications for machine learning practice.

Abstract

The problem of attempting to learn the mapping between data and labels is the crux of any machine learning task. It is, therefore, of interest to the machine learning community on practical as well as theoretical counts to consider the existence of a test or criterion for deciding the feasibility of attempting to learn. We investigate the existence of such a criterion in the setting of PAC-learning, basing the feasibility solely on whether the mapping to be learnt lends itself to approximation by a given class of hypothesis functions. We show that no such criterion exists, exposing a fundamental limitation in the decidability of learning. In other words, we prove that testing for PAC-learnability is undecidable in the Turing sense. We also briefly discuss some of the probable implications of this result to the current practice of machine learning.