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Discretized Langevin diffusions are efficient Monte Carlo methods forsampling from high dimensional target densities that are log-Lipschitz-smoothand (strongly) log-concave. In particular, the Euclidean Langevin Monte Carlosampling algorithm has received much attention lately, leading to a detailedunderstanding of its non-asymptotic convergence properties and of the role thatsmoothness and log-concavity play in the convergence rate. Distributions thatdo not possess these regularity properties can be addressed by considering aRiemannian Langevin diffusion with a metric capturing the local geometry of thelog-density. However, the Monte Carlo algorithms derived from discretizationsof such Riemannian Langevin diffusions are notoriously difficult to analyze. Inthis paper, we consider Langevin diffusions on a Hessian-type manifold andstudy a discretization that is closely related to the mirror-descent scheme. Weestablish for the first time a non-asymptotic upper-bound on the sampling errorof the resulting Hessian Riemannian Langevin Monte Carlo algorithm. This boundis measured according to a Wasserstein distance induced by a Riemannian metricground cost capturing the Hessian structure and closely related to aself-concordance-like condition. The upper-bound implies, for instance, thatthe iterates contract toward a Wasserstein ball around the target density whoseradius is made explicit. Our theory recovers existing Euclidean results and cancope with a wide variety of Hessian metrics related to highly non-flatgeometries.

Wasserstein Control of Mirror Langevin Monte Carlo