反向扩散的一阶平稳性理论
First-Order Stationarity of Reverse Diffusions
一篇把扩散采样和非凸优化联系起来的理论论文，证明了反向 SDE 流在强凸势下的指数收缩速率，做扩散模型理论方向的朋友可以看看。
arXiv 论文《First-Order Stationarity of Reverse Diffusions》为扩散模型建立了对应的一阶理论。论文证明，只要前向过程的平稳势满足强凸性，过阻尼与欠阻尼 Langevin 扩散的 SDE 反向时间流就会以显式指数速率收缩相对 Fisher 散度。作者指出该条件只作用于所选加噪过程，而不作用于数据，且这是 SDE 反向扩散独有的优势，基于 ODE 的反向过程不具备。论文进一步引入离散化分析，为过阻尼与欠阻尼扩散采样器建立了平均一阶平稳性界，对应非凸优化中的平均梯度范数保证。该凸性无关的证书是局部的，只保证 score 一致性，不保证全局模式权重。
First-Order Stationarity of Reverse Diffusions
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.
