Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling

• URL: http://arxiv.org/abs/2407.16936v1
• Date: Wed, 24 Jul 2024 02:15:48 GMT
• Title: Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling
• Authors: Wei Guo, Molei Tao, Yongxin Chen,
• Abstract summary: We establish an oracle complexity of $widetildeOleft(fracdbeta2cal A2varepsilon6right)$ for simple annealed Langevin Monte Carlo algorithm. We show that $cal A$ represents the action of a curve of probability measures interpolating the target distribution $pi$ and a readily sampleable distribution.
• Score: 28.931489333515618
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We address the outstanding problem of sampling from an unnormalized density that may be non-log-concave and multimodal. To enhance the performance of simple Markov chain Monte Carlo (MCMC) methods, techniques of annealing type have been widely used. However, quantitative theoretical guarantees of these techniques are under-explored. This study takes a first step toward providing a non-asymptotic analysis of annealed MCMC. Specifically, we establish, for the first time, an oracle complexity of $\widetilde{O}\left(\frac{d\beta^2{\cal A}^2}{\varepsilon^6}\right)$ for simple annealed Langevin Monte Carlo algorithm to achieve $\varepsilon^2$ accuracy in Kullback-Leibler divergence to the target distribution $\pi\propto{\rm e}^{-V}$ on $\mathbb{R}^d$ with $\beta$-smooth potential $V$. Here, ${\cal A}$ represents the action of a curve of probability measures interpolating the target distribution $\pi$ and a readily sampleable distribution.

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