The Mirror Langevin Algorithm Converges with Vanishing Bias
Abstract
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).
Cite
@article{arxiv.2109.12077,
title = {The Mirror Langevin Algorithm Converges with Vanishing Bias},
author = {Ruilin Li and Molei Tao and Santosh S. Vempala and Andre Wibisono},
journal= {arXiv preprint arXiv:2109.12077},
year = {2021}
}