# From stability of Langevin diffusion to convergence of proximal MCMC for non-log-concave sampling

arXiv: https://arxiv.org/abs/2505.14177 · HTML: https://mlsanity.com/paper/2505.14177/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 May 2025.

## Built on by (most cited first, 6 of 6)

- [FlowLPS: Langevin-Proximal Sampling for Flow-based Inverse Problem Solvers](https://mlsanity.com/paper/2512.07150/) (2025)
- [DC-LA: Difference-of-Convex Langevin Algorithm](https://mlsanity.com/paper/2601.22932/) (2026)
- [Outlier-robust Diffusion Posterior Sampling for Bayesian Inverse Problems](https://mlsanity.com/paper/2602.02045/) (2026)
- [Stochastic Momentum Methods for Non-smooth Non-Convex Finite-Sum Coupled Compositional Optimization](https://mlsanity.com/paper/2506.02504/) (2025)
- [From the Gradient-Step Denoiser to the Proximal Denoiser and their associated convergent Plug-and-Play algorithms](https://mlsanity.com/paper/2509.09793/) (2025)
- [Proximal-Based Generative Modeling for Bayesian Inverse Problems](https://mlsanity.com/paper/2605.13278/) (2026)

## Abstract (arXiv)

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.