A variational approach to dimension-free self-normalized concentration

TL;DR

This paper develops a variational (PAC-Bayes) approach to derive dimension-free self-normalized concentration inequalities for vector-valued stochastic processes, covering a broad class of tail behaviors and generalizing existing bounds.

Contribution

It introduces a novel variational method to obtain dimension-free self-normalized concentration bounds for sub-ψ processes, extending previous results and filling gaps between determinant-based and condition number-based bounds.

Findings

Generalized sub-ψ concentration bounds including heavy-tailed cases

Proved a Bernstein inequality for vectors with moment conditions

First dimension-free self-normalized empirical Bernstein inequality

Abstract

We study the self-normalized concentration of vector-valued stochastic processes. We focus on bounds for "sub-$\psi$" processes, a well-known and quite general class of process that encompasses a wide variety of well-known tail conditions (including sub-exponential, sub-Gaussian, sub-gamma, sub-Poisson, and several heavy-tailed settings without a moment generating function such as symmetric or bounded 2nd or 3rd moments). Our results recover and generalize the influential bound of de la Pe\~na et al. [20] (proved again in Abbasi-Yadkori et al. [2]) in the sub-Gaussian case. Further, we fill a gap in the literature between determinant-based bounds and more recent bounds based on condition numbers. As applications we prove a Bernstein inequality for random vectors satisfying a moment condition (a more general condition than boundedness), and also provide the first dimension-free self-normalized empirical Bernstein inequality. Our techniques are based on the variational (PAC-Bayes) approach to concentration.