PAC-Bayes Generalization bounds for Score Based Diffusion Models

Score-based diffusion models are known to generalize well from finite data, yet there exists no formal theoretical guarantee. To date, no generalization bounds exist that connect the empirical score-matching loss to the true generation error in terms of the number of training samples ($n$) and the discretization time step ($\Delta t$) under general assumptions. This paper provides the first PAC-Bayesian generalization bounds for diffusion models, upper-bounding the Wasserstein-1 distance between the true and generated distributions. We employ the PAC-Bayesian framework because, unlike traditional complexity measures that often scale poorly and become vacuous in high dimensions, it can provide sharper, posterior-specific, and often dimension-free bounds that have proven effective in deep learning. Our framework is robust, remaining valid for practical settings such as non-smooth data distributions, without any requirement for early stopping. For the general case of unbounded scores, we establish a generalization error rate of $O(n^{-1/8})$, requiring a sampling time step of $\Delta t = O(n^{-5/8})$. These results provide a principled guide for selecting training samples and offer a new theoretical lens for understanding generalization in diffusion models.