Sqrt(d) Dimension Dependence of Langevin Monte Carlo
Sqrt(d) Dimension Dependence of Langevin Monte Carlo
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发表时间：
2021-09
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通讯作者：
Ruilin Li;H. Zha;Molei Tao
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作者：
Ruilin Li;H. Zha;Molei Tao
本文考虑了流行的MCMC方法的未调整的朗之万蒙特卡罗（LMC），并提供了一个非渐近分析的抽样误差在2-Wasserstein距离。该证明基于Li et al.（2019）中的均方分析的改进，并且这种改进的框架自动分析了基于收缩性SDEs离散化的一大类采样算法。使用这个框架，我们建立了一个$\tilde{O}（\sqrt{d}/\sqrt）$混合时间界的LMC，没有热启动，在常见的对数光滑和对数强凸的条件下，加上一个增长条件的三阶导数的潜在的目标措施。该界改进了先前已知的最佳结果，并且对于满足上述假设的目标测量，在维度d和精度公差方面都是最优的（就阶而言）。数值实验进一步验证了我们的理论分析。
This article considers the popular MCMC method of unadjusted Langevin Monte Carlo (LMC) and provides a non-asymptotic analysis of its sampling error in 2-Wasserstein distance. The proof is based on a refinement of mean-square analysis in Li et al. (2019), and this refined framework automates the analysis of a large class of sampling algorithms based on discretizations of contractive SDEs. Using this framework, we establish an $\tilde{O}(\sqrt{d}/\epsilon)$ mixing time bound for LMC, without warm start, under the common log-smooth and log-strongly-convex conditions, plus a growth condition on the 3rd-order derivative of the potential of target measures. This bound improves the best previously known $\tilde{O}(d/\epsilon)$ result and is optimal (in terms of order) in both dimension $d$ and accuracy tolerance $\epsilon$ for target measures satisfying the aforementioned assumptions. Our theoretical analysis is further validated by numerical experiments.