无凸性Schr{<$}dinger-F{<$}llmer采样器的收敛性分析
Convergence Analysis of Schr{ö}dinger-F{ö}llmer Sampler without Convexity
摘要：
Schrödinger-Föllmer采样器（SFS）是一种新颖而有效的方法，用于从可能未归一化的分布中采样，而没有遍历性。SFS是基于单位区间上的Schrödinger-Föllmer扩散过程$$\mathrm{d} X_{t}=-\nabla U\left(X_t, t\right) \mathrm{d} t+\mathrm{d} B_{t}, \quad t \in[0,1],\quad X_0=0$$的Euler-Maruyama离散化，将时间0的退化分布转化为时间1的目标分布.在\cite{sfs21}中，SFS的一致性是在一个限制性假设下建立的，即漂移项$b(x,t)$和势$U(x,t)$的%一致（在$t$上）是强%凹凸（在$x$上）。本文在目标分布与标准正态分布的密度比为光滑有界的条件下，给出了SFS在瓦瑟斯坦距离下的非渐近误差界，但不要求势函数是强凸的.
英文摘要：
Schrödinger-Föllmer 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 Schrödinger-Föllmer 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.
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