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).

该研究探讨了优化中将问题的几何形状从欧几里得度量变为黑塞度量的技术，并研究了基于镜面流的镜面 Langevin 算法的偏差收敛性和混合时间界限。研究表明，这种算法可以通过自协调条件来实现收敛。