Polynomial complexity sampling from multimodal distributions using Sequential Monte Carlo

2025-08-03

AI Summary

This paper addresses the problem of efficiently sampling from the Gibbs measure associated with a nonconvex energy function at low temperatures, where conventional MCMC methods suffer from exponentially slow global mixing due to multimodality. We propose a hierarchical sequential Monte Carlo framework that integrates geometric annealing scheduling with continuous-time Langevin diffusion; crucially, it only requires local mixing within energy basins at each temperature level. Theoretically, we prove that this strategy reduces the total sampling time complexity to polynomial order—specifically, (O(eta^4)) in inverse temperature (eta) and (O(varepsilon^{-2})) in target accuracy (varepsilon)—thereby circumventing the exponential dependence on global mixing. We further derive an explicit error bound and empirically validate the algorithm’s efficacy and convergence rate on canonical nonconvex models.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• 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
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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 forth 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.

Problem

• Sampling from multimodal distributions efficiently
• Analyzing Sequential Monte Carlo with geometric annealing
• Estimating convergence with polynomial time complexity

Innovation

• Sequential Monte Carlo with geometric annealing
• Langevin diffusion for local mixing
• Polynomial complexity with temperature dependence