On Bernstein Type Exponential Inequalities for Matrix Martingales

TL;DR

This paper extends Bernstein's concentration inequalities to matrix-valued discrete-time martingales, providing new bounds using Lieb's theory and supermartingale techniques, thus broadening the scope of matrix concentration results.

Contribution

It introduces a novel exponential supermartingale construction for matrix martingales, extending Tropp's inequalities with a new proof approach.

Findings

Derived Bernstein-type inequalities for matrix martingales

Established a new supermartingale construction based on Lieb's theory

Extended existing matrix concentration bounds to broader settings

Abstract

In this work, Bernstein's concentration inequalities for squared integrable matrix-valued discrete-time martingales are obtained. Based on Lieb's theory and Bernstein's condition, a suitable supermartingale can be constructed. Our proof is largely based on this new exponential supermartingale, Freedman's method, and Doob's stopping theorem. Our result can be regarded as an extension of Tropp's work (ECP, 2012).