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We consider the following statistical estimation problem: given a family F ofreal valued functions over some domain X and an i.i.d. sample drawn from anunknown distribution P over X, find h in F such that the expectation of hw.r.t. P is probably approximately equal to the supremum over expectations onmembers of F. This Expectation Maximization (EMX) problem captures many wellstudied learning problems; in fact, it is equivalent to Vapnik's generalsetting of learning.  Surprisingly, we show that the EMX learnability, as well as the learningrates of some basic class F, depend on the cardinality of the continuum and istherefore independent of the set theory ZFC axioms (that are widely accepted asa formalization of the notion of a mathematical proof).  We focus on the case where the functions in F are Boolean, which generalizesclassification problems. We study the interaction between the statisticalsample complexity of F and its combinatorial structure. We introduce a newversion of sample compression schemes and show that it characterizes EMXlearnability for a wide family of classes. However, we show that for the classof finite subsets of the real line, the existence of such compression schemesis independent of set theory. We conclude that the learnability of that classwith respect to the family of probability distributions of countable support isindependent of the set theory ZFC axioms.  We also explore the existence of a "VC-dimension-like" parameter thatcaptures learnability in this setting. Our results imply that that there existno "finitary" combinatorial parameter that characterizes EMX learnability in away similar to the VC-dimension based characterization of binary valuedclassification problems.

A learning problem that is independent of the set theory ZFC axioms