Exact rates in Vapnik–Chervonenkis bounds

Abstract

Vapnik–Chervonenkis bounds on rates of uniform convergence of empirical means to their expectations have been continuously improved over the years since the precursory work in [26]. The result obtained by Talagrand in 1994 [21] seems to provide the final word as far as universal bounds are concerned. However, in the case where there are some additional assumptions on the underlying probability distribution, the exponential rate of convergence can be fairly improved. Alexander [1] and Massart [15] have found better exponential rates (similar to those in Bennett–Bernstein inequalities) under the assumption of a control on the variance of the empirical process. In this paper, the case of a particular distribution is considered for the empirical process indexed by a family of sets, and we provide the exact exponential rate based on large deviations theorems, as predicted by Azencott [2].