MASLA: Metropolis-Adjusted Subdifferential Langevin Sampling for Nonsmooth Nonconvex Potentials

Sampling from distributions with nonsmooth nonconvex potentials challenges gradient-based Langevin methods and can complicate proximal approaches. We introduce Metropolis-adjusted subdifferential Langevin algorithm (MASLA), which combines a fixed measurable selection from a conservative field with an exact Metropolis–Hastings correction. The method accommodates locally Lipschitz potentials without requiring smoothing or a proximal mapping. We establish well-posedness, reversibility, and exact invariance of the target distribution, together with positive Harris recurrence. Our central theoretical result rigorously proves geometric ergodicity for positive-definite quadratic potentials perturbed by Lipschitz functions admitting bounded conservative fields. The result provides an explicit admissible range of step sizes and holds from every initial point, including points of nondifferentiability. Numerical experiments verify the predicted geometric convergence behavior on a convex composite target and demonstrate competitive performance against proximal and subgradient samplers. Further experiments on a nonsmooth nonconvex target show that the Metropolis correction substantially reduces sampling error relative to unadjusted subdifferential Langevin dynamics. These results establish MASLA as a theoretically grounded sampling method for a class of nonsmooth nonconvex distributions.