Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm

• URL: http://arxiv.org/abs/2012.12810v1
• Date: Wed, 23 Dec 2020 17:14:06 GMT
• Title: Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm
• Authors: Sinho Chewi, Chen Lu, Kwangjun Ahn, Xiang Cheng, Thibaut Le Gouic, Philippe Rigollet
• Abstract summary: Mixing time of MALA on class of log-smooth and strongly log-concave distributions is $O(d)$. New technique based on projection characterization of the Metropolis adjustment reduces study of MALA to the well-studied discretization analysis of the Langevin SDE.
• Score: 22.19906823739798
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Conventional wisdom in the sampling literature, backed by a popular diffusion scaling limit, suggests that the mixing time of the Metropolis-Adjusted Langevin Algorithm (MALA) scales as $O(d^{1/3})$, where $d$ is the dimension. However, the diffusion scaling limit requires stringent assumptions on the target distribution and is asymptotic in nature. In contrast, the best known non-asymptotic mixing time bound for MALA on the class of log-smooth and strongly log-concave distributions is $O(d)$. In this work, we establish that the mixing time of MALA on this class of target distributions is $\widetilde\Theta(d^{1/2})$ under a warm start. Our upper bound proof introduces a new technique based on a projection characterization of the Metropolis adjustment which reduces the study of MALA to the well-studied discretization analysis of the Langevin SDE and bypasses direct computation of the acceptance probability.

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