Spectral gap of nonreversible Markov chains

TL;DR

This paper introduces a generalized spectral gap for nonreversible Markov chains, linking it to convergence times and showing that empirical averages can converge faster than the chain's mixing time.

Contribution

It defines the spectral gap for nonreversible chains, relates relaxation time to empirical convergence, and extends classical inequalities to this broader setting.

Findings

Relaxation time can be characterized by empirical average convergence.

Empirical averages often converge faster than the mixing time.

Inequalities relate spectral gap, Cheeger constant, and mixing time.

Abstract

We define the spectral gap of a Markov chain on a finite state space as the second-smallest singular value of the generator of the chain, generalizing the usual definition of spectral gap for reversible chains. We then define the relaxation time of the chain as the inverse of this spectral gap, and show that this relaxation time can be characterized, for any Markov chain, as the time required for convergence of empirical averages. This relaxation time is related to the Cheeger constant and the mixing time of the chain through inequalities that are similar to the reversible case, and the path argument can be used to get upper bounds. Several examples are worked out. An interesting finding from the examples is that the time for convergence of empirical averages in nonreversible chains can often be substantially smaller than the mixing time.