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# Mixing Time Estimation in Ergodic Markov Chains from a Single Trajectory with Contraction Methods

*Proceedings of the 31st International Conference on Algorithmic Learning Theory*, PMLR 117:890-905, 2020.

#### Abstract

The mixing time $t_{\mathsf{mix}}$ of an ergodic Markov chain measures the rate of convergence towards its stationary distribution $\boldsymbol{\pi}$. We consider the problem of estimating $t_{\mathsf{mix}}$ from one single trajectory of $m$ observations $(X_1, …, X_m)$, in the case where the transition kernel $\boldsymbol{M}$ is unknown, a research program started by Hsu et al. [2015]. The community has so far focused primarily on leveraging spectral methods to estimate the

*relaxation time*$t_{\mathsf{rel}}$ of a*reversible*Markov chain as a proxy for $t_{\mathsf{mix}}$. Although these techniques have recently been extended to tackle non-reversible chains, this general setting remains much less understood. Our new approach based on contraction methods is the first that aims at directly estimating $t_{\mathsf{mix}}$ up to multiplicative small universal constants instead of $t_{\mathsf{rel}}$. It does so by introducing a generalized version of Dobrushin’s contraction coefficient $\kappa_{\mathsf{gen}}$, which is shown to control the mixing time regardless of reversibility. We subsequently design fully data-dependent high confidence intervals around $\kappa_{\mathsf{gen}}$ that generally yield better convergence guarantees and are more practical than state-of-the-art.