随机微分方程终值分布估计的拆分方法

中文摘要 AI 辅助

在涉及随机微分方程的许多场景中，包括基于扩散的生成式人工智能，我们的目标是从终值分布中准确生成样本。通常，这是通过生成扩散路径的独立同分布样本来实现的。在给定固定模拟预算的情况下，一种合理的提高效率的方法可能是通过适当拆分部分路径来生成路径树。这暗示了性能的提升，但人们担心引入的依赖性。在本文中，我们全面研究了这个问题。以Kolmogorov-Smirnov距离作为精度度量，我们确定了当模拟预算趋于无穷大时相关经验分布的极限误差。我们刻画了一种由相应渐近优化问题驱动的拆分策略。理论结果揭示了问题中优雅的底层结构。实际实现包括两个阶段：初始估计阶段和最终推断阶段。总体而言，在许多设置中，我们观察到平均误差相对于独立同分布样本有10-25%的提升。在一项探索性的CIFAR-10研究中，我们的方法将最大平均差异降低了8-13%。

英文摘要

In many settings involving stochastic differential equations, including in diffusion based generative AI, our aim is to accurately generate samples from a terminal distribution. Typically, this is done by generating i.i.d. samples of diffusion paths. 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. With Kolmogorov-Smirnov distance as a measure of accuracy, 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. The theoretical results bring out the elegant underlying structure in the problem. Practical implementation involves two phases, an initial estimation phase and a final inference phase. Overall, we observe a 10-25% improvement in mean error over i.i.d. samples in many settings. In an exploratory CIFAR-10 study, our method reduces the maximum mean discrepancy by 8-13%.

发表机构

• Ashoka University（阿育王大学）

机构由 AI 辅助整理，请以论文原文为准。