Diffusion models are a class of generative models that serve to establish a stochastic transport map between an empirically observed, yet unknown, target distribution and a known prior. Despite their remarkable success in real-world applications, a theoretical understanding of their generalization capabilities remains underdeveloped. This work embarks on a comprehensive theoretical exploration of the generalization attributes of diffusion models. We establish theoretical estimates of the generalization gap that evolves in tandem with the training dynamics of score-based diffusion models, suggesting a polynomially small generalization error ($O(n^{-2/5}+m^{-4/5})$) on both the sample size $n$ and the model capacity $m$, evading the curse of dimensionality (i.e., not exponentially large in the data dimension) when early-stopped. Furthermore, we extend our quantitative analysis to a data-dependent scenario, wherein target distributions are portrayed as a succession of densities with progressively increasing distances between modes. This precisely elucidates the adverse effect of "modes shift" in ground truths on the model generalization. Moreover, these estimates are not solely theoretical constructs but have also been confirmed through numerical simulations. Our findings contribute to the rigorous understanding of diffusion models' generalization properties and provide insights that may guide practical applications.

翻译：扩散模型是一类生成模型，旨在建立经验观测但未知的目标分布与已知先验分布之间的随机传输映射。尽管这类模型在现实应用中取得了显著成功，但其泛化能力的理论理解仍不完善。本文对扩散模型的泛化属性展开了全面的理论探索。我们建立了与基于分数的扩散模型训练动态同步演化的泛化差距理论估计，表明当采用早停策略时，其泛化误差关于样本量n和模型容量m呈多项式小量级（$O(n^{-2/5}+m^{-4/5})$），从而规避了维度灾难（即不随数据维度呈指数增长）。此外，我们将定量分析扩展至数据依赖场景，其中目标分布被刻画为一系列模式间距离递增的密度分布，这精确阐明了真实分布中“模式偏移”对模型泛化的不利影响。值得注意的是，这些估计并非纯理论构建，也已通过数值模拟得到验证。我们的研究为深入理解扩散模型的泛化特性提供了严格的理论基础，并可为实际应用提供指导性见解。