标题： 扩散模型中的隐式正则化：一种算法依赖的泛化分析 显示英文标题

标题： Implicit Regularisation in Diffusion Models: An Algorithm-Dependent Generalisation Analysis

摘要： 去噪扩散模型的成功引发了关于其泛化行为的重要问题，尤其是在高维设置中。值得注意的是，当训练和采样完美进行时，这些模型会记忆训练数据——这意味着某种形式的正则化对于泛化是必不可少的。现有的理论分析主要依赖于算法无关的技术，如统一收敛，大量利用模型结构来获得泛化界限。在本工作中，我们则利用促进扩散模型泛化的算法方面，为这种设置开发了一个算法相关的泛化理论。借鉴算法稳定性的框架，我们引入了分数稳定性这一概念，它量化了分数匹配算法对数据集扰动的敏感性。我们根据分数稳定性推导出泛化界限，并将我们的框架应用于几个基本的学习设置，识别正则化来源。特别是，我们考虑了带有提前停止的去噪分数匹配（去噪正则化）、采样器范围内的粗粒度离散化（采样器正则化）以及使用SGD优化（优化正则化）。通过将我们的分析建立在算法属性而非模型结构上，我们识别出了一些扩散模型特有的隐式正则化来源，这些在文献中迄今一直被忽视。 显示英文摘要

摘要： The success of denoising diffusion models raises important questions regarding their generalisation behaviour, particularly in high-dimensional settings. Notably, it has been shown that when training and sampling are performed perfectly, these models memorise training data -- implying that some form of regularisation is essential for generalisation. Existing theoretical analyses primarily rely on algorithm-independent techniques such as uniform convergence, heavily utilising model structure to obtain generalisation bounds. In this work, we instead leverage the algorithmic aspects that promote generalisation in diffusion models, developing a general theory of algorithm-dependent generalisation for this setting. Borrowing from the framework of algorithmic stability, we introduce the notion of score stability, which quantifies the sensitivity of score-matching algorithms to dataset perturbations. We derive generalisation bounds in terms of score stability, and apply our framework to several fundamental learning settings, identifying sources of regularisation. In particular, we consider denoising score matching with early stopping (denoising regularisation), sampler-wide coarse discretisation (sampler regularisation) and optimising with SGD (optimisation regularisation). By grounding our analysis in algorithmic properties rather than model structure, we identify multiple sources of implicit regularisation unique to diffusion models that have so far been overlooked in the literature.