{"ID":2884851,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.06483","arxiv_id":"2508.06483","title":"A variational approach to dimension-free self-normalized concentration","abstract":"We study the self-normalized concentration of vector-valued stochastic processes. We focus on bounds for \"sub-$ψ$\" 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 Peña 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.","short_abstract":"We study the self-normalized concentration of vector-valued stochastic processes. We focus on bounds for \"sub-$ψ$\" 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-...","url_abs":"https://arxiv.org/abs/2508.06483","url_pdf":"https://arxiv.org/pdf/2508.06483v2","authors":"[\"Ben Chugg\",\"Aaditya Ramdas\"]","published":"2025-08-08T17:44:09Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.ST\",\"stat.ML\"]","methods":"[]","has_code":false}