Multimodal sampling via Schr\"odinger-F\"ollmer samplers with temperatures

TL;DR

This paper introduces temperature-parameterized Schr"odinger-F"ollmer samplers with improved convergence rates for sampling from complex distributions, especially multimodal ones, outperforming Langevin methods.

Contribution

The paper develops a new temperature-based Schr"odinger-F"ollmer sampler with enhanced convergence analysis, achieving an order ${ m O}(h)$ rate, and demonstrates its effectiveness over Langevin samplers.

Findings

High temperatures improve multimodal sampling.

Enhanced convergence rate of order ${ m O}(h)$ under smoothness conditions.

SFS outperforms Langevin samplers in experiments.

Abstract

Generating samples from complex and high-dimensional distributions is ubiquitous in various scientific fields of statistical physics, Bayesian inference, scientific computing and machine learning. Very recently, Huang et al. (IEEE Trans. Inform. Theory, 2025) proposed new Schr\"odinger-F\"ollmer samplers (SFS), based on the Euler discretization of the Schr\"odinger-F\"ollmer diffusion evolving on the unit interval $[0, 1]$. There, a convergence rate of order $\mathcal{O}(\sqrt{h})$ in the $L^2$-Wasserstein distance was obtained for the Euler discretization with a uniform time step-size $h>0$. By incorporating a temperature parameter, different samplers are introduced in this paper, based on the Euler discretization of the Schr\"odinger-F\"ollmer process with temperatures. As revealed by numerical experiments, high temperatures are vital, particularly in sampling from multimodal distributions. Further, a novel approach of error analysis is developed for the time discretization and an enhanced convergence rate of order ${ \mathcal{O}(h)}$ is obtained in the $L^2$-Wasserstein distance, under certain smoothness conditions on the drift. This significantly improves the existing order-half convergence in the aforementioned paper. Unlike Langevin samplers, SFS is of gradient-free, works in a unit interval $[0, 1]$ and does not require any ergodicity. Numerical experiments confirm the convergence rate and show that, the SFS substantially outperforms vanilla Langevin samplers, particularly in sampling from multimodal distributions.