  The spectral gap $\gamma$ 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 $\gamma$ from this data. Let $\pi$
be the stationary distribution of $P$, and $\pi_\star = \min_x \pi(x)$. They
showed that, if $t = \tilde{O}\bigl(\frac{1}{\gamma^3 \pi_\star}\bigr)$, then
$\gamma$ can be estimated to within multiplicative constants with high
probability. They also proved that $\tilde{\Omega}\bigl(\frac{n}{\gamma}\bigr)$
steps are required for precise estimation of $\gamma$. We show that
$\tilde{O}\bigl(\frac{1}{\gamma \pi_\star}\bigr)$ steps of the chain suffice to
estimate $\gamma$ up to multiplicative constants with high probability. When
$\pi$ is uniform, this matches (up to logarithmic corrections) the lower bound
of Hsu, Kontorovich, and Szepesvari.
