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。

镜子 Langevin 算法在消失偏差时收敛

The Mirror Langevin Algorithm Converges with Vanishing Bias

Estimating mutual information from i.i.d. samples drawn from an unknown joint
density function is a basic statistical problem of broad interest with
multitudinous applications. The most popular estimator is one proposed by
Kraskov and St\"ogbauer and Grassberger (KSG) in 2004, and is nonparametric and
based on the distances of each sample to its $k^{\rm th}$ nearest neighboring
sample, where $k$ is a fixed small integer. Despite its widespread use (part of
scientific software packages), theoretical properties of this estimator have
been largely unexplored. In this paper we demonstrate that the estimator is
consistent and also identify an upper bound on the rate of convergence of the
bias as a function of number of samples. We argue that the superior performance
benefits of the KSG estimator stems from a curious "correlation boosting"
effect and build on this intuition to modify the KSG estimator in novel ways to
construct a superior estimator. As a byproduct of our investigations, we obtain
nearly tight rates of convergence of the $\ell_2$ error of the well known fixed
$k$ nearest neighbor estimator of differential entropy by Kozachenko and
Leonenko.

本文研究了基于 $KSG$ 估计的互信息估计中，样本数对偏差收敛速度的影响，发现了 $KSG$ 估计器的优越性能来源于 “相关性提升” 效应，并通过改进 $KSG$ 估计器构建出更优秀的估计器。