论文标题
镜像扩散的指数千古性
Exponential ergodicity of mirror-Langevin diffusions
论文作者
论文摘要
由Zhang等人在镜像 - 朗格文素扩散中进行清洁的非质合收敛分析，这是由未经条件的对数凸出分布进行采样的问题的动机。 （2020）。作为此框架的特殊情况，我们提出了一类称为牛顿 - 兰格文散射的扩散，并证明它们会以不仅尺寸不含尺寸的速率，而且对目标分布不依赖，从而呈指数级的平稳性。我们将此结果应用于使用内部方法启发的策略中从凸体上均匀分布进行采样的问题。我们的一般方法遵循联系采样和优化的最新趋势，并突出了卡方差异的作用。特别是，它会产生有关瓦斯坦距离兰格文鸟扩散的收敛性的新结果。
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 highlights the role of the chi-squared divergence. In particular, it yields new results on the convergence of the vanilla Langevin diffusion in Wasserstein distance.
