Sample compression schemes for VC classes

Sample compression schemes were defined by Littlestone and Warmuth (1986) as an abstraction of the structure underlying many learning algorithms. Roughly speaking, a sample compression scheme of size κ means that given an arbitrary list of labeled examples, one can retain only κ of them in a way that allows us to recover the labels of all other examples in the list. They showed that compression implies probably approximately correct learnability for binary-labeled classes and asked whether the other direction holds. We answer their question and show that every concept class C with VC dimension d has a sample compression scheme of size exponential in d.