证明了非凸势函数下近端随机梯度Langevin算法的收敛性，为图像重建等难题提供理论支持。
From stability of Langevin diffusion to convergence of proximal MCMC for non-log-concave sampling
- 基于Moreau包络和U/LA稳定性分析，构建非凸势函数下的收敛框架
- 在合成数据和图像反问题中验证，近端方法收敛更快且保持重建精度
- 首次给出非凸势函数下近端SG-Langevin算法的严格收敛证明
研究从非凸势函数中采样的问题，聚焦于无调整Langevin算法（ULA）。在势函数无穷远处强凸的假设下，证明了离散时间ULA对漂移近似的稳定性。在图像反问题等场景中，势函数常为非凸且非光滑。近端随机梯度Langevin算法（PSGLA）结合前向-后向优化与ULA步骤，是处理此类问题的主流方法。本文通过主要稳定性结果与Moreau包络性质，首次推导出非凸势函数下PSGLA的收敛性证明。在合成数据及图像反问题中进行了实验验证，结果显示：相比标准SG-Langevin算法，PSGLA在后验采样中表现出更快速的收敛速度，同时保持了良好的恢复性能。
原文摘要 · Abstract (English)
We consider the problem of sampling distributions stemming from non-convex potentials with Unadjusted Langevin Algorithm (ULA). We prove the stability of the discrete-time ULA to drift approximations under the assumption that the potential is strongly convex at infinity. In many context, e.g. imaging inverse problems, potentials are non-convex and non-smooth. Proximal Stochastic Gradient Langevin Algorithm (PSGLA) is a popular algorithm to handle such potentials. It combines the forward-backward optimization algorithm with a ULA step. Our main stability result combined with properties of the Moreau envelope allows us to derive the first proof of convergence of the PSGLA for non-convex potentials. We empirically validate our methodology on synthetic data and in the context of imaging inverse problems. In particular, we observe that PSGLA exhibits faster convergence rates than Stochastic Gradient Langevin Algorithm for posterior sampling while preserving its restoration properties.
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