Undecidability of Learnability

Machine learning researchers and practitioners steadily enlarge the multitude of successful learning models. They achieve this through in-depth theoretical analyses and experiential heuristics. However, there is no known general-purpose procedure for rigorously evaluating whether newly proposed models indeed successfully learn from data. We show that such a procedure cannot exist. For PAC binary classification, uniform and universal online learning, and exact learning through teacher-learner interactions, learnability is in general undecidable, both in the sense of independence of the axioms in a formal system and in the sense of uncomputability. Our proofs proceed via computable constructions of function classes that encode the consistency problem for formal systems and the halting problem for Turing machines into complexity measures that characterize learnability. Our work shows that undecidability appears in the theoretical foundations of machine learning: There is no one-size-fits-all algorithm for deciding whether a machine learning model can be successful. We cannot in general automatize the process of assessing new learning models.