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Machine learning researchers and practitioners steadily enlarge the multitudeof successful learning models. They achieve this through in-depth theoreticalanalyses and experiential heuristics. However, there is no knowngeneral-purpose procedure for rigorously evaluating whether newly proposedmodels indeed successfully learn from data. We show that such a procedurecannot exist. For PAC binary classification, uniform and universal onlinelearning, and exact learning through teacher-learner interactions, learnabilityis in general undecidable, both in the sense of independence of the axioms in aformal system and in the sense of uncomputability. Our proofs proceed viacomputable constructions that encode the consistency problem for formal systemsand the halting problem for Turing machines into whether certain functionclasses are trivial/finite or highly complex, which we then relate to whetherthese classes are learnable via established characterizations of learnabilitythrough complexity measures. Our work shows that undecidability appears in thetheoretical foundations of artificial intelligence: There is noone-size-fits-all algorithm for deciding whether a machine learning model canbe successful. We cannot in general automatize the process of assessing newlearning models.

From Undecidability of Non-Triviality and Finiteness to Undecidability of Learnability