There has been a surge of works bridging MCMC sampling and optimization, with a specific focus on translating non-asymptotic convergence guarantees for optimization problems into the analysis of Langevin algorithms in MCMC sampling. A conspicuous distinction between the convergence analysis of Langevin sampling and that of optimization is that all known convergence rates for Langevin algorithms depend on the dimensionality of the problem, whereas the convergence rates for optimization are dimension-free for convex problems. Whether a dimension independent convergence rate can be achieved by Langevin algorithm is thus a long-standing open problem. This paper provides an affirmative answer to this problem for large classes of either Lipschitz or smooth convex problems with normal priors. By viewing Langevin algorithm as composite optimization, we develop a new analysis technique that leads to dimension independent convergence rates for such problems.

翻译：连接MCMC取样和优化的工程激增,特别侧重于将非非不方便的优化融合保证转化为对MCMC取样的Langevin算法的分析。 Langevin取样和优化的趋同分析明显区别在于,Langevin算法所有已知的趋同率取决于问题的方方面面,而优化的趋同率对于交汇问题是没有维度的。因此,Langevin算法能否实现维度独立的趋同率是一个长期存在的未决问题。本文为大类的Libschitz或平滑的与正常前期的convex问题提供了这一问题的肯定答案。通过将Langevin算法视为综合优化,我们开发了一种新的分析技术,从而导致这些问题的维度独立趋同率。