Motivated by the problem of sampling from ill-conditioned log-concave distributions, we give a clean non-asymptotic convergence analysis of mirror-Langevin diffusions as introduced in Zhang et al. (2020). As a special case of this framework, we propose a class of diffusions called Newton-Langevin diffusions and prove that they converge to stationarity exponentially fast with a rate which not only is dimension-free, but also has no dependence on the target distribution. We give an application of this result to the problem of sampling from the uniform distribution on a convex body using a strategy inspired by interior-point methods. Our general approach follows the recent trend of linking sampling and optimization, and in particular, it yields new results on the convergence of the vanilla Langevin diffusion in Wasserstein distance.

研究了一种从不合适条件的对数凹分布中进行采样的方法，证明了其具有无关维数和目标分布的快速收敛速率，进一步应用于优化中，提供了一种基于内点法的策略来从凸体上分布中进行均匀分布采样，这一方法新颖性的体现在于指出了 chi-squared divergence 的作用，为朴素 Langevin 漂移在 Wasserstein 距离内的收敛提供了新的结果。

镜像朗之万扩散的指数遍历性