Concentration inequalities for Markov chains by Marton couplings and spectral methods

TL;DR

This paper extends concentration inequalities to Markov chains, introducing a pseudo spectral gap for non-reversible chains and providing bounds on empirical averages using coupling and spectral methods.

Contribution

It develops new concentration inequalities for Markov chains, including a pseudo spectral gap concept for non-reversible chains, combining coupling and spectral techniques.

Findings

Proves McDiarmid's inequality for Markov chains with mixing time dependence

Establishes variance bounds and Bernstein inequalities for empirical averages

Introduces the pseudo spectral gap for non-reversible chains

Abstract

We prove a version of McDiarmid's bounded differences inequality for Markov chains, with constants proportional to the mixing time of the chain. We also show variance bounds and Bernstein-type inequalities for empirical averages of Markov chains. In the case of non-reversible chains, we introduce a new quantity called the "pseudo spectral gap", and show that it plays a similar role for non-reversible chains as the spectral gap plays for reversible chains. Our techniques for proving these results are based on a coupling construction of Katalin Marton, and on spectral techniques due to Pascal Lezaud. The pseudo spectral gap generalises the multiplicative reversiblication approach of Jim Fill.