Weak transport inequalities and applications to exponential inequalities and oracle inequalities

TL;DR

This paper extends weak transport inequalities beyond the Hamming distance, applying them to Euclidean norms and time series, leading to new concentration and oracle inequalities with fast convergence rates.

Contribution

It introduces dimension-free weak transport inequalities for Euclidean norms, applicable to non-product measures and dependent data, enhancing concentration and oracle inequalities.

Findings

Extended Talagrand inequalities to Euclidean norms

Derived new concentration inequalities for time series

Established oracle inequalities with fast convergence rates

Abstract

We extend the dimension free Talagrand inequalities for convex distance \cite{talagrand:1995} using an extension of Marton's weak transport \cite{marton:1996a} to other metrics than the Hamming distance. We study the dual form of these weak transport inequalities for the euclidian norm and prove that it implies sub-gaussianity and convex Poincar\'e inequality \cite{bobkov:gotze:1999a}. We obtain new weak transport inequalities for non products measures extending the results of Samson in \cite{samson:2000}. Many examples are provided to show that the euclidian norm is an appropriate metric for classical time series. Our approach, based on trajectories coupling, is more efficient to obtain dimension free concentration than existing contractive assumptions \cite{djellout:guillin:wu:2004,marton:2004}. Expressing the concentration properties of the ordinary least square estimator as a conditional weak transport problem, we derive new oracle inequalities with fast rates of convergence in dependent settings.