  The Hidden Markov Model (HMM) is one of the mainstays of statistical modeling
of discrete time series, with applications including speech recognition,
computational biology, computer vision and econometrics. Estimating an HMM from
its observation process is often addressed via the Baum-Welch algorithm, which
is known to be susceptible to local optima. In this paper, we first give a
general characterization of the basin of attraction associated with any global
optimum of the population likelihood. By exploiting this characterization, we
provide non-asymptotic finite sample guarantees on the Baum-Welch updates,
guaranteeing geometric convergence to a small ball of radius on the order of
the minimax rate around a global optimum. As a concrete example, we prove a
linear rate of convergence for a hidden Markov mixture of two isotropic
Gaussians given a suitable mean separation and an initialization within a ball
of large radius around (one of) the true parameters. To our knowledge, these
are the first rigorous local convergence guarantees to global optima for the
Baum-Welch algorithm in a setting where the likelihood function is nonconvex.
We complement our theoretical results with thorough numerical simulations
studying the convergence of the Baum-Welch algorithm and illustrating the
accuracy of our predictions.
