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。尽管这类模型在实际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进行了全面的理论探索。我们建立了基于评分函数的扩散模型在训练动态过程中泛化差距的理论估计，表明当采用早停策略时，泛化误差随样本量$n$和模型容量$m$呈多项式小量级（$O(n^{-2/5}+m^{-4/5})$），且规避了维度灾难（即不随数据维度呈指数增长）。此外，我们将定量分析扩展到数据依赖场景，其中目标分布被刻画为一系列密度函数，其模态间距逐步增大。这精确揭示了真实分布中“模态偏移”对模型泛化的不利影响。值得注意的是，这些估计不仅是理论构建，还通过数值模拟得到了验证。本研究为深入理解扩散模型的泛化性质提供了理论依据，并为实际应用提供了指导性见解。