A variational approach to dimension-free self-normalized concentration
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Chugg, Ben Ramdas, Aaditya |
| author_facet | Chugg, Ben Ramdas, Aaditya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06483 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A variational approach to dimension-free self-normalized concentration Chugg, Ben Ramdas, Aaditya Probability Statistics Theory Machine Learning 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. |
| title | A variational approach to dimension-free self-normalized concentration |
| topic | Probability Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2508.06483 |