A Vector Bernstein Inequality for Self-Normalized Martingales

Published: 27 Mar 2025, Last Modified: 27 Mar 2025Accepted by TMLREveryoneRevisionsBibTeXCC BY 4.0
Abstract: We prove a Bernstein inequality for vector-valued self-normalized martingales. We first give an alternative perspective of the corresponding sub-Gaussian bound due to Abbasi-Yadkori et al. via a PAC-Bayesian argument with Gaussian priors. By instantiating this argument to priors drawn uniformly over well-chosen ellipsoids, we obtain a Bernstein bound.
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: * Fixed a few typos (thanks!) Changes completed per request by Reviewer gDu1: * added a short note on how to obtain equation (7). * Broke away subsection 1.1 into its own section (now sec 2). * added references: Freedman/De la Pena (bernstein-type bounds) & McAllester / Shawe-Taylor (PAC-Bayes) * defined the variance proxy in section 2.2 Changes completed per request by Reviewer 4FTh: * Added an additional example/explanation as to the use of the result as a refined confidence ellipsoid for least squares estimation
Assigned Action Editor: ~Jasper_C.H._Lee1
Submission Number: 3852
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