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

TL;DR

This work tackles efficient sampling from non-log-concave, multimodal targets π ∝ e^{-V} by developing annealed Langevin diffusion and Monte Carlo (ALD/ALMC). It introduces a curve of intermediate distributions with finite action and uses Girsanov-based analysis to bound the global KL error, achieving a non-asymptotic complexity of \\\widetilde{O}(d β^2 {\cal A}^2 / ε^6) to reach KL accuracy ε^2, without relying on log-concavity or isoperimetric conditions. The discretization leverages an exponential-integrator scheme and a single LMC step per intermediate distribution, with a carefully chosen partition to control discretization error. Experiments on Gaussian mixtures illustrate polynomial dependence of complexity on problem parameters, contrasting with exponential lower bounds under strong isoperimetric requirements. Overall, the paper broadens the toolkit for non-convex sampling by providing rigorous non-asymptotic guarantees for annealed MCMC with practical interpolation schemes.

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 $\widetilde{O}\left(\frac{dβ^2{\cal A}^2}{\varepsilon^6}\right)$ for the simple annealed Langevin Monte Carlo algorithm to achieve $\varepsilon^2$ accuracy in Kullback-Leibler divergence to the target distribution $π\propto{\rm e}^{-V}$ on $\mathbb{R}^d$ with $β$-smooth potential $V$. Here, ${\cal A}$ represents the action of a curve of probability measures interpolating the target distribution $π$ and a readily sampleable distribution.

Figures (2)

• Figure 1: Illustration of the ALMC algorithm. The $\ell$-th green arrow, proceeding from left to right, represents one step of LMC towards $\widetilde{\pi}_{T_\ell}$ with step size $h_\ell$, while each red arrow corresponds to the application of the same transition kernel, initialized at $\widetilde{\pi}_{T_{\ell-1}}$ on the reference trajectory $\mathbb{P}$, which is depicted in purple. To evaluate $\operatorname{KL}(\mathbb{P}\|\mathbb{Q})$, the Girsanov theorem implies that we only need to bound the aggregate KL divergence across each small interval (i.e., the sum of the blue "distances").
• Figure 2: Relationship between norm of means $r$ and the number of iterations to reach $0.2$ and $0.1$ accuracy in KL divergence, both in $\log$ scale.

Theorems & Definitions (18)

• Lemma 1
• Lemma 2
• Lemma 3
• Example 1
• Theorem 1
• proof
• Theorem 2
• Example 2
• Lemma 4: Sufficient Condition for Absolute Continuity (Informal)
• proof