Improved Estimation of Relaxation Time in Non-reversible Markov Chains

TL;DR

This paper establishes the optimal sample complexity for estimating the pseudo-spectral gap in ergodic Markov chains, improves empirical methods with adaptive procedures, and introduces new theoretical tools like reversible dilation.

Contribution

It provides the first tight bounds for pseudo-spectral gap estimation, enhances empirical algorithms with adaptivity, and introduces the concept of reversible dilation of stochastic matrices.

Findings

Optimal sample complexity of rac{1}{b3_{ps} \u03c0_*} for estimating the pseudo-spectral gap.

An improved, fully adaptive empirical procedure with better confidence intervals.

Introduction of the reversible dilation concept for stochastic matrices.

Abstract

We show that the minimax sample complexity for estimating the pseudo-spectral gap $\gamma_{\mathsf{ps}}$ of an ergodic Markov chain in constant multiplicative error is of the order of $$\tilde{\Theta}\left( \frac{1}{\gamma_{\mathsf{ps}} \pi_{\star}} \right),$$ where $\pi_\star$ is the minimum stationary probability, recovering the known bound in the reversible setting for estimating the absolute spectral gap [Hsu et al., 2019], and resolving an open problem of Wolfer and Kontorovich [2019]. Furthermore, we strengthen the known empirical procedure by making it fully-adaptive to the data, thinning the confidence intervals and reducing the computational complexity. Along the way, we derive new properties of the pseudo-spectral gap and introduce the notion of a reversible dilation of a stochastic matrix.