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

Wei Guo, Molei Tao, Yongxin Chen

Keywords: MCMC, Annealed Langevin Monte Carlo, Non-log-concave sampling, Non-asymptotic analysis

TL;DR: We obtain the first non-asymptotic complexity analysis of annealed Langevin Monte Carlo for sampling non-log-concave distributions.

Abstract: We consider 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 the 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.

Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

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Submission Number: 7795