Convergence of Langevin AIS for multimodal distributions

TL;DR

This paper analyzes the convergence rates of Langevin Annealed Importance Sampling for multimodal distributions, showing quadratic time complexity in inverse temperature and providing bounds based on spectral estimates.

Contribution

It introduces bounds on the convergence of Langevin AIS for multimodal Gibbs measures, linking error control to spectral properties and inverse temperature.

Findings

Time complexity is quadratic in inverse temperature for fixed error thresholds.

A simple quantity controlling sampling error is identified and bounded.

Bounds are extended to an autonormalized version of the algorithm.

Abstract

We study convergence rates of the annealed importance sampling algorithm (Neal '01) combined with Langevin Monte Carlo when the target is a multimodal Gibbs measure. The main result shows that for a fixed error threshold, the time complexity is quadratic in the inverse temperature. We identify a simple and useful quantity that controls the sampling error for AIS in a general setting, and then bound this quantity in our setting using spectral estimates. We also study an autonormalized version and obtain bounds for the time complexity in terms of the inverse temperature.