Schr{\"o}dinger-F{\"o}llmer Sampler: Sampling without Ergodicity

TL;DR

The Schrödinger-Föllmer sampler (SFS) offers a novel, ergodicity-free method for sampling from complex distributions using a diffusion process, with theoretical guarantees and superior empirical performance.

Contribution

This paper introduces SFS, a new sampling method based on a Schrödinger-Föllmer diffusion process that does not require ergodicity, unlike traditional MCMC methods.

Findings

SFS provides better sample quality than existing methods.

Non-asymptotic error bounds are established for SFS.

SFS is easy to implement with Euler-Maruyama discretization.

Abstract

Sampling from probability distributions is an important problem in statistics and machine learning, specially in Bayesian inference when integration with respect to posterior distribution is intractable and sampling from the posterior is the only viable option for inference. In this paper, we propose Schr\"{o}dinger-F\"{o}llmer sampler (SFS), a novel approach for sampling from possibly unnormalized distributions. The proposed SFS is based on the Schr\"{o}dinger-F\"{o}llmer diffusion process on the unit interval with a time dependent drift term, which transports the degenerate distribution at time zero to the target distribution at time one. Comparing with the existing Markov chain Monte Carlo samplers that require ergodicity, no such requirement is needed for SFS. Computationally, SFS can be easily implemented using the Euler-Maruyama discretization. In theoretical analysis, we establish non-asymptotic error bounds for the sampling distribution of SFS in the Wasserstein distance under suitable conditions. We conduct numerical experiments to evaluate the performance of SFS and demonstrate that it is able to generate samples with better quality than several existing methods.