Variational Tail Bounds for Norms of Random Vectors and Matrices
Sohail Bahmani
TL;DR
This work develops variational tail bounds for norms of random vectors by expressing the target norm as a supremum over a dual ball and applying a simple, tractable coupling framework. The authors introduce a main lemma that yields moment and tail bounds via a variational choice of an aggregating distribution P_0, and then instantiate it with a pushforward Gaussian construction to obtain explicit, calculable bounds. They demonstrate that the approach reproduces classical Gaussian bounds for both polyhedral and Euclidean norms, and extend it to concentration results for sums of random PSD matrices and for sample covariance matrices under sub-exponential marginals, connecting to existing dimension-free bounds in the literature. A broader coupling abstraction is also provided to apply the method to a wider class of functionals beyond inner products. The results offer a flexible, interpretable framework for non-asymptotic norm concentration with practical parameter choices.
Abstract
We propose a variational tail bound for norms of random vectors under moment assumptions on their one-dimensional marginals. We also propose a simplified version of the bound that parametrizes the ``aggregating'' distribution in the proposed variational bound by considering a certain pushforward of the Gaussian distribution. We show that the proposed method reproduces some of the well-known bounds on norms of Gaussian random vectors, as well as a recent dimension-free concentration inequality for the spectral norm of sum of independent and identically distributed positive semidefinite matrices with sub-exponential marginals. We also obtain a similar concentration inequality for the sample covariance matrix of sub-exponential random vectors. Furthermore, we use coupling to formulate an abstraction of the proposed approach that applies more broadly.