Convergence Analysis of Schr{\"o}dinger-F{\"o}llmer Sampler without Convexity

TL;DR

This paper provides a nonasymptotic error bound for the Schr"odinger-F"ollmer sampler (SFS) in Wasserstein distance, removing the previous requirement of strong convexity of the potential, under certain smoothness conditions.

Contribution

It extends the convergence analysis of SFS to non-convex potentials by establishing error bounds without the strong convexity assumption.

Findings

Nonasymptotic Wasserstein error bounds for SFS

Convergence results under smooth, bounded density ratio conditions

Removal of the strong convexity requirement for the potential

Abstract

Schr\"{o}dinger-F\"{o}llmer sampler (SFS) is a novel and efficient approach for sampling from possibly unnormalized distributions without ergodicity. SFS is based on the Euler-Maruyama discretization of Schr\"{o}dinger-F\"{o}llmer diffusion process $$\mathrm{d} X_{t}=-\nabla U\left(X_t, t\right) \mathrm{d} t+\mathrm{d} B_{t}, \quad t \in[0,1],\quad X_0=0$$ on the unit interval, which transports the degenerate distribution at time zero to the target distribution at time one. In \cite{sfs21}, the consistency of SFS is established under a restricted assumption that %the drift term $b(x,t)$ the potential $U(x,t)$ is uniformly (on $t$) strongly %concave convex (on $x$). In this paper we provide a nonasymptotic error bound of SFS in Wasserstein distance under some smooth and bounded conditions on the density ratio of the target distribution over the standard normal distribution, but without requiring the strongly convexity of the potential.