A Refinement of Vapnik--Chervonenkis' Theorem
This work provides a theoretical refinement for machine learning researchers, offering incremental improvements to a foundational theorem.
The authors tackled the problem of refining the Vapnik-Chervonenkis theorem by improving the rate of uniform convergence for empirical probabilities, achieving a moderate-deviation sharpening with an additional factor of order (ε√n)^{-1} in the leading exponential term when ε√n is large.
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.