Multimodal sampling via Schrödinger-Föllmer samplers with temperatures
For researchers in Bayesian inference and scientific computing needing efficient sampling from multimodal distributions, this work provides a gradient-free sampler with provably faster convergence and practical advantages over Langevin methods.
The paper introduces temperature-parameterized Schrödinger-Föllmer samplers (SFS) for multimodal sampling, achieving an improved convergence rate of O(h) in L^2-Wasserstein distance compared to the previous O(√h), and demonstrates through numerical experiments that high-temperature SFS significantly outperforms vanilla Langevin samplers on multimodal distributions.
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 Schrödinger-Föllmer samplers (SFS), based on the Euler discretization of the Schrödinger-Föllmer 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 Schrödinger-Föllmer 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.