Learnability is a Compact Property

Recent work on learning has yielded a striking result: the learnability of various problems can be undecidable, or independent of the standard ZFC axioms of set theory. Furthermore, the learnability of such problems can fail to be a property of finite character: informally, it cannot be detected by examining finite projections of the problem. On the other hand, learning theory abounds with notions of dimension that characterize learning and consider only finite restrictions of the problem, i.e., are properties of finite character. How can these results be reconciled? More precisely, which classes of learning problems are vulnerable to logical undecidability, and which are within the grasp of finite characterizations? We demonstrate that the difficulty of supervised learning with metric losses admits a tight finite characterization. In particular, we prove that the sample complexity of learning a hypothesis class can be detected by examining its finite projections. For realizable and agnostic learning with respect to a wide class of proper loss functions, we demonstrate an exact compactness result: a class is learnable with a given sample complexity precisely when the same is true of all its finite projections. For realizable learning with improper loss functions, we show that exact compactness of sample complexity can fail, and provide matching upper and lower bounds of a factor of 2 on the extent to which such sample complexities can differ. We conjecture that larger gaps are possible for the agnostic case. At the heart of our technical work is a compactness result concerning assignments of variables that maintain a class of functions below a target value, which generalizes Hall's classic matching theorem and may be of independent interest.