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arXiv · 2607.11719

A uniform relative deviation inequality for VC-subgraph classes

François Portier
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Abstract

We establish a new Bernstein-type deviation inequality for classes of functions whose complexity is characterized through subgraphs. The inequality is non-asymptotic, involves explicit constants, and features a relative normalization by the probability level. Applied to kernel density estimation, it produces a location and bandwidth-adaptive error bound between the estimator and the smoothed density, holding simultaneously over all points on the real line and all positive bandwidths. The proof is elementary, combining a new symmetrization principle, which incorporates the relative normalization, with the maximal sub-Gaussian inequality, and requires neither concentration nor entropy-integral arguments. When specialized to classes of sets, our technique improves the constants in the classical Vapnik-Chervonenkis inequality with relative deviation of Anthony and Shawe-Taylor (1993), reducing the factor 4 S A (2n) to S A (2n) in the right-tail inequality and to 3 S A (2n) in the left-tail inequality.

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François Portier. 2026-07-13. A uniform relative deviation inequality for VC-subgraph classes. https://arxiv.org/abs/2607.11719

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