Exponential ergodicity of mirror-Langevin diffusions
Exponential ergodicity of mirror-Langevin diffusions
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发表时间：
2020-05
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通讯作者：
Sinho Chewi;Thibaut Le Gouic;Chen Lu;Tyler Maunu;P. Rigollet;Austin J. Stromme
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作者：
Sinho Chewi;Thibaut Le Gouic;Chen Lu;Tyler Maunu;P. Rigollet;Austin J. Stromme
受病态对数凹分布的采样问题的启发，我们给出了Zhang et al.（2020）中介绍的镜像朗之万扩散的一个干净的非渐近收敛分析。作为这个框架的一个特殊情况下，我们提出了一类称为牛顿-朗之万扩散，并证明了他们收敛到平稳指数快速的速度，不仅是无量纲的，而且也没有依赖于目标分布。我们给出了一个应用程序的结果，从凸体上的均匀分布的采样问题，使用的策略启发的邻域点方法。我们的一般方法遵循最近的趋势，链接采样和优化，并突出了卡方分歧的作用。特别是，它产生了新的结果的收敛香草Langevin扩散Wasserstein距离。
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.