PAC learnability of a concept class under non-atomic measures: a problem by Vidyasagar

Abstract

In response to a 1997 problem of M. Vidyasagar, we state a necessary and sufficient condition for distribution-free PAC learnability of a concept class $\mathcal{C}$ under the family of all non-atomic (diffuse) measures on the domain $\Omega$. Clearly, finiteness of the classical Vapnik-Chervonenkis dimension of $\mathcal{C}$ is a sufficient, but no longer necessary, condition. Besides, learnability of $\mathcal{C}$ under non-atomic measures does not imply the uniform Glivenko-Cantelli property with regard to non-atomic measures. Our learnability criterion is stated in terms of a combinatorial parameter $\VC({\mathscr C}\,{\mathrm{mod}}\,ω_1)$ which we call the VC dimension of $\mathcal{C}$ modulo countable sets. The new parameter is obtained by ``thickening up'' single points in the definition of VC dimension to uncountable ``clusters''. Equivalently, $\VC(\mathscr C\moddω_1)\leq d$ if and only if every countable subclass of $\mathcal{C}$ has VC dimension $\le d$ outside a countable subset of $\Omega$. The new parameter can be also expressed as the classical VC dimension of $\mathcal{C}$ calculated on a suitable subset of a compactification of $\Omega$. We do not make any measurability assumptions on $\mathcal{C}$, assuming instead the validity of Martin's Axiom (MA).

14 pages, 1 figure, latex 2e with Springer macros