Estimating the Mixing Time of Ergodic Markov Chains

TL;DR

This paper develops a method to estimate the mixing time of ergodic Markov chains from a single trajectory, overcoming challenges in the non-reversible case by focusing on the pseudo-spectral gap, and provides empirical confidence intervals with improved rates.

Contribution

It introduces a novel approach to estimate the pseudo-spectral gap for non-reversible chains, enabling polynomial dependence on key parameters and nearly minimax rates for mixing time estimation.

Findings

Achieves polynomial dependence on minimal stationary probability and pseudo-spectral gap.

Provides empirical confidence intervals shrinking at a rate of approximately 1/√m.

Improves estimation accuracy and theoretical guarantees over previous reversible-only methods.

Abstract

We address the problem of estimating the mixing time $t_{\mathsf{mix}}$ of an arbitrary ergodic finite-state Markov chain from a single trajectory of length $m$. The reversible case was addressed by Hsu et al. [2019], who left the general case as an open problem. In the reversible case, the analysis is greatly facilitated by the fact that the Markov operator is self-adjoint, and Weyl's inequality allows for a dimension-free perturbation analysis of the empirical eigenvalues. As Hsu et al. point out, in the absence of reversibility (which induces asymmetric pair probabilities matrices), the existing perturbation analysis has a worst-case exponential dependence on the number of states $d$. Furthermore, even if an eigenvalue perturbation analysis with better dependence on $d$ were available, in the non-reversible case the connection between the spectral gap and the mixing time is not nearly as straightforward as in the reversible case. Our key insight is to estimate the pseudo-spectral gap $\gamma_{\mathsf{ps}}$ instead, which allows us to overcome the loss of symmetry and to achieve a polynomial dependence on the minimal stationary probability $\pi_\star$ and $\gamma_{\mathsf{ps}}$. Additionally, in the reversible case, we obtain simultaneous nearly (up to logarithmic factors) minimax rates in $t_{\mathsf{mix}}$ and precision $\varepsilon$, closing a gap in Hsu et al., who treated $\varepsilon$ as constant in the lower bounds. Finally, we construct fully empirical confidence intervals for $\gamma_{\mathsf{ps}}$, which shrink to zero at a rate of roughly $1/\sqrt{m}$, and improve the state of the art in even the reversible case.