Convergence Analysis of Schr{\"o}dinger-F{\"o}llmer Sampler without Convexity
Abstract: Schr\"{o}dinger-F\"{o}llmer sampler (SFS) (Huang et al., 2021) 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 Huang et al. (2021), the consistency of SFS is established under a restricted assumption that the potential $U(x,t)$ is uniformly (on $t$) strongly convex (on $x$). In this paper we provide a non-asymptotic error bound of SFS in Wasserstein-2 distance under some smooth and bounded conditions on the density ratio of the target distribution over the standard normal distribution, but without requiring strong convexity of the potential.
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: This version is a revised one following from the comments of reviewers, and it is of higher quality.
We hope that we have adequately addressed the comments of reviewers.
Assigned Action Editor: ~Alain_Durmus1
Submission Number: 130
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