The VC dimension of partial concept classes via Radon's theorem

Abstract

Following Alon, Hanneke, Holzman, and Moran (FOCS 2021), we define a partial concept class (PCC) as a family of partial functions \(f: V\to\{0,1,\ast\}\); equivalently, its concepts partition the ground set into black ($f^{-1}(1)$), grey ($f^{-1}(\ast)$), and white parts ($f^{-1}(0)$). Its VC dimension is defined by shattering sets on which the value $\ast$ is not taken. We study two geometric PCCs in real Banach spaces, both with a margin \(δ>0\): expanded half-spaces, where the grey part is a strip of width at least \(δ\) adjacent to a half-space, and expanded balls, where the grey part is an annulus of width \(δ\) around a unit radius ball. Our main results are dimension-free upper bounds on the VC dimension of the PCC of expanded balls in \(L_p\parenthμ\), \(1\le p<\infty\), including the non-Euclidean and algorithmically particularly relevant case \(\ell^d_1\). These bounds depend on the margin and on the radii, but not on the ambient dimension or the underlying measure space. These are extensions of the work of Bourneuf, Charbit, and Thomassé (FOCS 2025) who studied the PCC of expanded balls in Euclidean space, that is, $\ell_2^d$. We also prove lower bounds on the VC dimension that match the upper bounds in terms of the margin parameter $δ$. Finally, we derive a Dense Neighborhood Lemma in \(L_p\)-spaces, again extending the known Euclidean results. Our method relies on the linearization of the distance through a map into a space of non-trivial Rademacher type, and then the use of a balanced signed-sum estimate, or a no-dimensional Radon theorem. The arguments rely on ideas from functional analysis that are clearly explained for the non-expert in that field.