  Discretized Langevin diffusions are efficient Monte Carlo methods for
sampling from high dimensional target densities that are log-Lipschitz-smooth
and (strongly) log-concave. In particular, the Euclidean Langevin Monte Carlo
sampling algorithm has received much attention lately, leading to a detailed
understanding of its non-asymptotic convergence properties and of the role that
smoothness and log-concavity play in the convergence rate. Distributions that
do not possess these regularity properties can be addressed by considering a
Riemannian Langevin diffusion with a metric capturing the local geometry of the
log-density. However, the Monte Carlo algorithms derived from discretizations
of such Riemannian Langevin diffusions are notoriously difficult to analyze. In
this paper, we consider Langevin diffusions on a Hessian-type manifold and
study a discretization that is closely related to the mirror-descent scheme. We
establish for the first time a non-asymptotic upper-bound on the sampling error
of the resulting Hessian Riemannian Langevin Monte Carlo algorithm. This bound
is measured according to a Wasserstein distance induced by a Riemannian metric
ground cost capturing the Hessian structure and closely related to a
self-concordance-like condition. The upper-bound implies, for instance, that
the iterates contract toward a Wasserstein ball around the target density whose
radius is made explicit. Our theory recovers existing Euclidean results and can
cope with a wide variety of Hessian metrics related to highly non-flat
geometries.
