📝 SummarXiv

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.