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Estimating the Spectral Gap of a Reversible Markov Chain from a Short Trajectory

The spectral gap $γ$ of an ergodic and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $t$ may be observed. Hsu, Kontorovich, and Szepesvari (2015) considered the problem of estimating $γ$ from this data. Let $π$ be the stationary distribution of $P$, and $π_\star = \min_x π(x)$. They showed that, if $t = \tilde{O}\bigl(\frac{1}{γ^3 π_\star}\bigr)$, then $γ$ can be estimated to within multiplicative constants with high probability. They also proved that $\tildeΩ\bigl(\frac{n}γ\bigr)$ steps are required for precise estimation of $γ$. We show that $\tilde{O}\bigl(\frac{1}{γπ_\star}\bigr)$ steps of the chain suffice to estimate $γ$ up to multiplicative constants with high probability. When $π$ is uniform, this matches (up to logarithmic corrections) the lower bound of Hsu, Kontorovich, and Szepesvari.