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

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 introducing a temperature parameter, different samplers are proposed 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 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.