Vector-valued self-normalized concentration inequalities beyond sub-Gaussianity
Diego Martinez-Taboada, Tomas Gonzalez, Aaditya Ramdas
TL;DR
The paper develops time-uniform concentration inequalities for vector-valued self-normalized processes in separable Hilbert spaces beyond sub-Gaussian tails by introducing a Pinelis-based nonnegative supermartingale that decouples direction from noise. It yields Bernstein- and Bennett-type bounds, including sequential and empirical variants, and ties the pseudo-variance to the information gain via the elliptical potential lemma, enabling dimension-free results. These inequalities are then translated into online inference tools for online linear regression and kernelized linear bandits, providing confidence ellipsoids and regret bounds that adapt to light-tailed noises and variance structure. The work broadens applicability of self-normalized concentration in infinite-dimensional settings and offers practical benefit for sequential decision-making under non-sub-Gaussian noise.
Abstract
The study of self-normalized processes plays a crucial role in a wide range of applications, from sequential decision-making to econometrics. While the behavior of self-normalized concentration has been widely investigated for scalar-valued processes, vector-valued processes remain comparatively underexplored, especially outside of the sub-Gaussian framework. In this contribution, we provide concentration bounds for self-normalized processes with light tails beyond sub-Gaussianity (such as Bennett or Bernstein bounds). We illustrate the relevance of our results in the context of online linear regression, with applications in (kernelized) linear bandits.