A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm
Chi Jin, Praneeth Netrapalli, Rong Ge, Sham M. Kakade, Michael I. Jordan
TL;DR
The paper introduces norm-subGaussian (nSG) random vectors, a framework that encompasses both subGaussian and norm-bounded vectors, and develops tight concentration bounds for sums and vector martingales of such vectors. It establishes equivalent characterizations in tails, moments, and MGFs, with projections remaining subGaussian and the norm-squared being subExponential. Employing Lieb's concavity theorem and a matrix-MGF approach, the authors derive high-probability bounds for sums of nSG vectors that depend only logarithmically on the dimension $d$, including a Hoeffding-type inequality and a two-case bound for random variance parameters. These results advance dimension-efficient concentration for high-dimensional vector sequences and highlight an open problem regarding the optimality of the logarithmic dimension dependence.
Abstract
In this note, we derive concentration inequalities for random vectors with subGaussian norm (a generalization of both subGaussian random vectors and norm bounded random vectors), which are tight up to logarithmic factors.