A Vector Bernstein Inequality for Self-Normalized Martingales
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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