逆扩散的一阶平稳性

中文摘要 AI 辅助

近期文献展示了优化与采样之间的紧密联系。我们为扩散模型建立了相应的一阶理论。首先，基于SDE的过阻尼和欠阻尼朗之万扩散的逆时流，在前向过程的平稳势强凸时——这是对所选择的加噪过程的条件，而非对数据的条件——以显式指数速率收缩相对Fisher散度。这是基于SDE的逆扩散的独特优势，在基于ODE的逆过程中不存在。其次，我们纳入离散化，为过阻尼和欠阻尼扩散模型的采样器建立了平均一阶平稳性界——这是非凸优化中平均梯度范数保证的采样类比。与非凸优化一样，无凸性证书是局部的：它保证得分一致性，而非全局模式权重。

英文摘要

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 models. As in nonconvex optimization, the convexity-free certificate is local: it guarantees score consistency, not global mode weights.

发表机构

• UW-Madison（威斯康星大学麦迪逊分校）

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