A variational approach to dimension-free self-normalized concentration

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.