Uniform concentration and symmetrization for weak interactions

TL;DR

This paper extends the use of Gaussian and Rademacher complexities to derive uniform bounds for nonlinear statistics, providing tight bounds for U-statistics, smoothened L-statistics, and error functionals in regularized algorithms.

Contribution

It introduces a novel method for deriving uniform bounds for nonlinear statistics, broadening the applicability of complexity measures in statistical learning.

Findings

Tight bounds for U-statistics established

Bounds for smoothened L-statistics derived

Error functionals of l2-regularized algorithms analyzed

Abstract

The method to derive uniform bounds with Gaussian and Rademacher complexities is extended to the case where the sample average is replaced by a nonlinear statistic. Tight bounds are obtained for U-statistics, smoothened L-statistics and error functionals of l2-regularized algorithms.