Almost tight bounds for ɛ-Nets

Abstract

Given any natural number d, 0 <1, let fd(e{open}) denote the smallest integer f such that every range space of Vapnik-Chervonenkis dimension d has an e{open}-net of size at most f. We solve a problem of Haussler and Welzl by showing that if d≥2, then {Mathematical expression} Further, we prove that f1(e{open})=max(2, {bottom right crop} 1/e{open} {bottom left crop}-1), and similar bounds are established for some special classes of range spaces of Vapnik-Chervonenkis dimension three. © 1992 Springer-Verlag New York Inc.