Improved estimation of relaxation time in nonreversible Markov chains

Abstract

We show that the minimax sample complexity for estimating the pseudo-spectral gap ${\mathit{\gamma }}_{\mathsf{ps}}$ of an ergodic Markov chain in constant multiplicative error is of the order of

$$\tilde{\mathrm{\Theta }}\left(\frac{1}{{\mathit{\gamma }}_{\mathsf{ps}}{\mathit{\pi }}_{\star }}\right),$$

where ${\mathit{\pi }}_{\star }$ is the minimum stationary probability, recovering the known bound in the reversible setting for estimating the absolute spectral gap (Hsu et al., Ann. Appl. Probab. 29 (2019) 2439–2480), and resolving an open problem of Wolfer and Kontorovich (In Proceedings of the Thirty-Second Conference on Learning Theory (2019) 3120–3159 PMLR). 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.