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

2024-07-24

arXiv.org

AI Summary

This work addresses the challenge of efficiently sampling from non-log-concave, multimodal, unnormalized densities. We propose Annealed Langevin Monte Carlo (ALMC), a novel sampling algorithm based on annealing. We establish the first non-asymptotic KL-divergence error bound for ALMC, introducing the *measure curve action* $mathcal{A}$ to quantify the geometric structure between the target and initial distributions. Under the assumption of $eta$-smooth nonconvex potentials, ALMC achieves $varepsilon^2$-KL accuracy. Its theoretical iteration complexity is $widetilde{mathcal{O}}(d eta^2 mathcal{A}^2 / varepsilon^6)$, substantially improving upon existing guarantees for standard Langevin methods in nonconvex settings. Our key contributions are: (i) the first oracle-level non-asymptotic convergence analysis for annealed Langevin MCMC; (ii) explicit incorporation of geometric structure—via $mathcal{A}$—into the sampling complexity; and (iii) a new, theoretically grounded paradigm for sampling multimodal distributions with provable finite-time guarantees.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $widetilde{O}left(frac{deta^2{cal A}^2}{varepsilon^6} ight)$ for the simple annealed Langevin Monte Carlo algorithm to achieve $varepsilon^2$ accuracy in Kullback-Leibler divergence to the target distribution $pipropto{ m e}^{-V}$ on $mathbb{R}^d$ with $eta$-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.

Problem

• Non-log-concave sampling challenge
• Annealed MCMC theoretical guarantees
• Langevin Monte Carlo oracle complexity

Innovation

• Annealed Langevin Monte Carlo
• Non-log-concave sampling
• Non-asymptotic analysis