Improving Generalization Bounds for VC Classes Using the Hypergeometric Tail Inversion

TL;DR

This paper introduces a novel approach to tighten generalization bounds for VC classes by leveraging hypergeometric tail inversion and optimizing the ghost sample trick, resulting in significantly improved, nearly non-vacuous bounds.

Contribution

The paper presents a new method combining hypergeometric tail inversion and ghost sample optimization to derive tighter, non-vacuous generalization bounds for VC classes.

Findings

Bounds are tighter than previous VC bounds across various data sizes.

The new bounds are nearly never vacuous in practical scenarios.

Derived bounds include relative deviation and multiclass margin bounds.

Abstract

We significantly improve the generalization bounds for VC classes by using two main ideas. First, we consider the hypergeometric tail inversion to obtain a very tight non-uniform distribution-independent risk upper bound for VC classes. Second, we optimize the ghost sample trick to obtain a further non-negligible gain. These improvements are then used to derive a relative deviation bound, a multiclass margin bound, as well as a lower bound. Numerical comparisons show that the new bound is nearly never vacuous, and is tighter than other VC bounds for all reasonable data set sizes.