Score-based generative models (SGMs) have emerged as one of the most popular classes of generative models. A substantial body of work now exists on the analysis of SGMs, focusing either on discretization aspects or on their statistical performance. In the latter case, bounds have been derived, under various metrics, between the true data distribution and the distribution induced by the SGM, often demonstrating polynomial convergence rates with respect to the number of training samples. However, these approaches adopt a largely approximation theory viewpoint, which tends to be overly pessimistic and relatively coarse. In particular, they fail to fully explain the empirical success of SGMs or capture the role of the optimization algorithm used in practice to train the score network. To support this observation, we first present simple experiments illustrating the concrete impact of optimization hyperparameters on the generalization ability of the generated distribution. Then, this paper aims to bridge this theoretical gap by providing the first algorithmic- and data-dependent generalization analysis for SGMs. In particular, we establish bounds that explicitly account for the optimization dynamics of the learning algorithm, offering new insights into the generalization behavior of SGMs. Our theoretical findings are supported by empirical results on several datasets.

翻译：基于分数的生成模型已成为最流行的生成模型类别之一。目前已有大量关于SGM分析的研究工作，主要关注离散化方面或其统计性能。在后一种情况下，研究者已在不同度量下推导出真实数据分布与SGM诱导分布之间的边界，通常证明了关于训练样本数量的多项式收敛速率。然而，这些方法主要采用近似理论视角，往往过于悲观且相对粗糙。特别是，它们未能充分解释SGM的经验成功，也未捕捉实践中用于训练分数网络的优化算法所起的作用。为支持这一观点，我们首先通过简单实验说明优化超参数对生成分布泛化能力的具体影响。随后，本文旨在通过为SGM提供首个算法与数据依赖的泛化分析来弥合这一理论空白。具体而言，我们建立的边界明确考虑了学习算法的优化动态，为SGM的泛化行为提供了新的见解。我们的理论发现在多个数据集上得到了实证结果的支持。