We first prove that Littlestone classes, those which model theorists call stable, characterize learnability in a new statistical model: a learner in this new setting outputs the same hypothesis, up to measure zero, with probability one, after a uniformly bounded number of revisions. This fills a certain gap in the literature, and sets the stage for an approximation theorem characterizing Littlestone classes in terms of a range of learning models, by analogy to definability of types in model theory. We then give a complete analogue of Shelah's celebrated (and perhaps a priori untranslatable) Unstable Formula Theorem in the learning setting, with algorithmic arguments taking the place of the infinite.
View on arXiv@article{malliaris2025_2212.05050,
title={ The unstable formula theorem revisited via algorithms },
author={ Maryanthe Malliaris and Shay Moran },
journal={arXiv preprint arXiv:2212.05050},
year={ 2025 }
}