A Refinement of Vapnik--Chervonenkis' Theorem
A. Iosevich, A. Vagharshakyan, E. Wyman
TL;DR
This work refines the Vapnik–Chervonenkis framework by sharpening the final probabilistic step of uniform convergence bounds. By replacing Hoeffding's inequality with a Berry–Esseen–controlled normal approximation, it achieves a moderate-deviation improvement in the tail bound for the empirical process, introducing an extra factor of order $(\varepsilon\sqrt{n})^{-1}$ in the leading term when $\varepsilon\sqrt{n}$ is large. The method preserves the VC structure via the growth function and VC dimension while delivering explicit bounds and constants. The results clarify the regime where the improvement is significant and open avenues for extensions to relative deviations and adaptive switching between bounds.
Abstract
Vapnik--Chervonenkis' theorem is a seminal result in machine learning. It establishes sufficient conditions for empirical probabilities to converge to theoretical probabilities, uniformly over families of events. It also provides an estimate for the rate of such uniform convergence. We revisit the probabilistic component of the classical argument. Instead of applying Hoeffding's inequality at the final step, we use a normal approximation with explicit Berry--Esseen error control. This yields a moderate-deviation sharpening of the usual VC estimate, with an additional factor of order $(\varepsilon\sqrt{n})^{-1}$ in the leading exponential term when $\varepsilon\sqrt{n}$ is large.