Optimal Bounds on the VC-dimension

TL;DR

This paper establishes optimal bounds on the VC-dimension for key set systems, resolving longstanding open problems and advancing understanding in machine learning and geometry.

Contribution

It provides the first tight bounds on VC-dimension for k-fold unions/intersections of half-spaces and the simplices set system, solving open problems from 1989.

Findings

Optimal bounds on VC-dimension for k-fold unions/intersections of half-spaces

Optimal bounds on VC-dimension for the simplices set system

Resolution of a 1989 open problem in machine learning

Abstract

The VC-dimension of a set system is a way to capture its complexity and has been a key parameter studied extensively in machine learning and geometry communities. In this paper, we resolve two longstanding open problems on bounding the VC-dimension of two fundamental set systems: $k$-fold unions/intersections of half-spaces, and the simplices set system. Among other implications, it settles an open question in machine learning that was first studied in the 1989 foundational paper of Blumer, Ehrenfeucht, Haussler and Warmuth as well as by Eisenstat and Angluin and Johnson.