Statistics Papers

Document Type

Journal Article

Date of this Version

12-2002

Publication Source

Journal of Computer and System Sciences

Volume

65

Issue

4

Start Page

660

Last Page

671

DOI

10.1016/S0022-0000(02)00022-3

Abstract

We study the complexity of approximating the VC dimension of a collection of sets, when the sets are encoded succinctly by a small circuit. We show that this problem is:

Σ3p-hard to approximate to within a factor 2−ε for all ε>0,
•approximable in AM to within a factor 2, and
•AM-hard to approximate to within a factor N1−ε for all ε>0.

To obtain the Σ3p-hardness result we solve a randomness extraction problem using list-decodable binary codes; for the positive result we utilize the Sauer–Shelah(–Perles) Lemma. We prove analogous results for the q-ary VC dimension, where the approximation threshold is q.

Copyright/Permission Statement

© 2002. This manuscript version is made available under the CC-BY-NC-ND 4.0 license.

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Date Posted: 27 November 2017

This document has been peer reviewed.