From stability of Langevin diffusion to convergence of proximal MCMC for non-log-concave sampling
Journal:
arXiv
Published Date:
May 20, 2025
Abstract
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.
