A Splitting Method for SDE Terminal-Law Estimation

In many settings involving stochastic differential equations (SDEs), including diffusion based generative AI, our aim is to accurately generate samples from a distribution. Typically, this is done by generating i.i.d. samples of the SDE. Given a fixed simulation budget, a reasonable way to gain efficiency may be to instead generate a tree of paths through appropriately split partial paths. This suggests improved performance, but one worries about the injected dependence. In this paper, we study this issue comprehensively. We develop a general method to construct the splitting design and analyze its performance for a broad class of error measures, giving insight into the elegant underlying structure in the problem. For the Kolmogorov-Smirnov (KS) distance and maximum mean discrepancy (MMD), we identify the limiting errors of the associated empirical distributions as the simulation budget increases to infinity. We characterize a splitting strategy motivated by a corresponding asymptotic optimization problem. Overall, we observe a 12-26% improvement in mean KS over i.i.d. samples in many settings. On CIFAR-10 and FFHQ, splitting lowers the MMD by 17-26% relative to i.i.d. samples.