  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.
