  We propose a new discretization of the mirror-Langevin diffusion and give a
crisp proof of its convergence. Our analysis uses relative convexity/smoothness
and self-concordance, ideas which originated in convex optimization, together
with a new result in optimal transport that generalizes the displacement
convexity of the entropy. Unlike prior works, our result both (1) requires much
weaker assumptions on the mirror map and the target distribution, and (2) has
vanishing bias as the step size tends to zero. In particular, for the task of
sampling from a log-concave distribution supported on a compact set, our
theoretical results are significantly better than the existing guarantees.
