Bernstein's inequalities for general Markov chains

TL;DR

This paper derives Bernstein's inequalities for functions of general Markov chains, providing sharp variance bounds and extending classical inequalities to dependent data, with applications in MCMC and robust mean estimation.

Contribution

It introduces Bernstein inequalities for general Markov chains, including non-reversible and state-space cases, with a novel convex analysis approach for dependence influence.

Findings

Achieves sharp variance proxies for Markov chains.

Provides tight deviation bounds in MCMC and mean estimation.

Extends classical Bernstein inequalities to dependent data.

Abstract

We establish Bernstein's inequalities for functions of general (general-state-space and possibly non-reversible) Markov chains. These inequalities achieve sharp variance proxies and encompass the classical Bernstein inequality for independent random variables as special cases. The key analysis lies in bounding the operator norm of a perturbed Markov transition kernel by the exponential of sum of two convex functions. One coincides with what delivers the classical Bernstein inequality, and the other reflects the influence of the Markov dependence. A convex analysis on these two functions then derives our Bernstein inequalities. As applications, we apply our Bernstein inequalities to the Markov chain Monte Carlo integral estimation problem and the robust mean estimation problem with Markov-dependent samples, and achieve tight deviation bounds that previous inequalities can not.