A Vector Bernstein Inequality for Self-Normalized Martingales

TL;DR

This paper establishes a Bernstein inequality for vector-valued self-normalized martingales, providing a new perspective and tighter bounds through a PAC-Bayesian approach with Gaussian priors.

Contribution

It introduces a novel Bernstein inequality for vector martingales using a PAC-Bayesian framework, offering an alternative to existing sub-Gaussian bounds.

Findings

Derived a Bernstein inequality for vector-valued self-normalized martingales.

Provided an alternative perspective on sub-Gaussian bounds via PAC-Bayesian methods.

Achieved tighter bounds by instantiating priors over ellipsoids.

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.