Time-complexity of sampling from a multimodal distribution using sequential Monte Carlo

TL;DR

This paper analyzes a sequential Monte Carlo method for sampling from complex, non-convex distributions at low temperatures, demonstrating that its time complexity scales polynomially with inverse temperature and error tolerance.

Contribution

It provides a theoretical analysis of the time complexity of a SMC algorithm using geometric annealing and Langevin diffusion for non-convex energy landscapes.

Findings

Monte Carlo estimators converge with time complexity scaling as the fourth power of inverse temperature.

Local mixing within energy valleys is sufficient, reducing overall sampling time.

Explicit estimates are provided in a simplified model scenario.

Abstract

We study a sequential Monte Carlo algorithm to sample from the Gibbs measure with a non-convex energy function at a low temperature. We use the practical and popular geometric annealing schedule, and use a Langevin diffusion at each temperature level. The Langevin diffusion only needs to run for a time that is long enough to ensure local mixing within energy valleys, which is much shorter than the time required for global mixing. Our main result shows convergence of Monte Carlo estimators with time complexity that, approximately, scales like the fourth power of the inverse temperature, and the square of the inverse allowed error. We also study this algorithm in an illustrative model scenario where more explicit estimates can be given.