Learnability Can Be Independent of Set Theory (Invited Paper)

Shai Ben-David, Pavel Hrubes, Shay Moran, Amir Shpilka, Amir Yehudayoff

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


A fundamental result in statistical learning theory is the equivalence of PAC learnability of a class with the finiteness of its Vapnik-Chervonenkis dimension. However, this clean result applies only to binary classification problems. In search for a similar combinatorial characterization of learnability in a more general setting, we discovered a surprising independence of set theory for some basic general notion of learnability. Consider the following statistical estimation problem: given a family F of real valued random variables over some domain X and an i.i.d. sample drawn from an unknown distribution P over X, find f in F such that its expectation w.r.t. P is close to the supremum expectation over all members of F. This Expectation Maximization (EMX) problem captures many well studied learning problems. Surprisingly, we show that the EMX learnability of some simple classes depends on the cardinality of the continuum and is therefore independent of the set theory ZFC axioms. Our results imply that that there exist no "finitary" combinatorial parameter that characterizes EMX learnability in a way similar to the VC-dimension characterization of binary classification learnability.
Original languageEnglish
Title of host publicationProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
Place of PublicationNew York, NY, USA
Number of pages1
StatePublished - 2021

Publication series

NameSTOC 2021
PublisherAssociation for Computing Machinery


  • Learnability


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