Bernstein-type and Bennett-type inequalities for unbounded matrix martingales

TL;DR

This paper develops explicit Bernstein-type and Bennett-type concentration inequalities for unbounded matrix martingales, extending classical results to the Hermitian matrix setting with applications to empirical and bounded difference inequalities.

Contribution

It introduces new concentration inequalities for matrix martingales with unbounded observations, incorporating effective rank and providing practical corollaries.

Findings

Derived Bernstein-type inequalities for matrix martingales.

Extended results to effective rank replacing ambient dimension.

Provided empirical Bernstein and McDiarmid's inequalities.

Abstract

We derive explicit Bernstein-type and Bennett-type concentration inequalities for matrix-valued martingale processes with unbounded observations from the Hermitian space $\mathbb{H}(d)$. Specifically, we assume that the $\psi_{\alpha}$-Orlicz (quasi-)norms of their difference process are bounded for some $\alpha > 0$. Further, we generalize the obtained result by replacing the ambient dimension $d$ with the effective rank of the covariance of the observations. To illustrate the applicability of the results, we prove several corollaries, including an empirical version of Bernstein's inequality and an extension of the bounded difference inequality, also known as McDiarmid's inequality.