[论文] First-Order Stationarity of Reverse Diffusions

论文概要

研究领域: ML 作者: Zhifeng Chen, Chenyang Jiang, Yazhen Wang 发布时间: 2026-09-25 arXiv: 2609.31612

中文摘要

近期文献揭示了优化与采样之间的深刻联系。我们为扩散模型建立了相应的一阶理论。第一，只要前向过程的平稳势函数强凸——这是对所选取噪过程的条件，而非对数据的条件——过阻尼与欠阻尼 Langevin 扩散基于 SDE 的逆时间流就能以显式的指数速率收缩相对 Fisher 散度。这是基于 SDE 的逆扩散独有的优势，在基于 ODE 的逆过程中并不存在。第二，我们引入离散化分析，为两类扩散模型的采样器建立了平均一阶平稳性界——即非凸优化中“平均梯度范数”保证的采样版本。与非凸优化中一样，这一无凸性条件的证书是局部的：它保证的是 score 的一致性，而非全局的模态权重。

原文摘要

Recent literature has shown a strong connection between optimization and sampling. We develop the corresponding first-order theory for diffusion models. First, the SDE-based reverse-time flows of overdamped and underdamped Langevin diffusions contract relative Fisher divergences at explicit exponential rates whenever the stationary potential of the forward process is strongly convex---a condition on the noising process one chooses, not on the data. This is a unique advantage of SDE-based reverse diffusion, absent in the reverse process based on ODEs. Second, we incorporate discretization and establish averaged first-order stationarity bounds---the sampling analog of averaged gradient-norm guarantees in nonconvex optimization---for samplers of both overdamped and underdamped diffusion model...

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