  The technique of modifying the geometry of a problem from Euclidean to
Hessian metric has proved to be quite effective in optimization, and has been
the subject of study for sampling. The Mirror Langevin Diffusion (MLD) is a
sampling analogue of mirror flow in continuous time, and it has nice
convergence properties under log-Sobolev or Poincare inequalities relative to
the Hessian metric, as shown by Chewi et al. (2020). In discrete time, a simple
discretization of MLD is the Mirror Langevin Algorithm (MLA) studied by Zhang
et al. (2020), who showed a biased convergence bound with a non-vanishing bias
term (does not go to zero as step size goes to zero). This raised the question
of whether we need a better analysis or a better discretization to achieve a
vanishing bias. Here we study the basic Mirror Langevin Algorithm and show it
indeed has a vanishing bias. We apply mean-square analysis based on Li et al.
(2019) and Li et al. (2021) to show the mixing time bound for MLA under the
modified self-concordance condition introduced by Zhang et al. (2020).
