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# Empirical bounds for functions with weak interactions

*Proceedings of the 31st Conference On Learning Theory*, PMLR 75:987-1010, 2018.

#### Abstract

We provide sharp empirical estimates of expectation, variance and normal approximation for a class of statistics whose variation in any argument does not change too much when another argument is modified. Examples of such weak interactions are furnished by U- and V-statistics, Lipschitz L-statistics and various error functionals of $\ell_2$-regularized algorithms and Gibbs algorithms.