Penalized Nonreversible Langevin for Constrained Sampling

Sampling from probability distributions restricted to a compact convex set C is a fundamental challenge in Bayesian inference and machine learning. Standard Langevin dynamics, which rely on unconstrained gradients, cannot be applied directly. This paper addresses this by combining two techniques: squared distance penalization, which replaces hard constraints with a smooth potential, and nonreversible Langevin dynamics, which introduces skew-symmetric perturbations to the drift to break detailed balance and accelerate mixing.

The authors propose a penalized nonreversible Langevin algorithm that evolves on the entire space, using the penalty gradient to enforce the constraint. They analyze both full-gradient and stochastic gradient variants. A key theoretical contribution is the construction of an adapted metric, P_J, based on the linearized nonreversible drift, which allows for a rigorous comparison between reversible and nonreversible convergence rates.

The authors establish nonasymptotic total variation and Wasserstein bounds for their proposed algorithms. For a two-dimensional quadratic model, they demonstrate that tuning the skew perturbation to the curvature imbalance—caused by the penalty parameter—removes the penalty-induced spectral stiffness. This tuning improves the iteration complexity from linear to logarithmic in the curvature ratio, providing a theoretical basis for nonreversible acceleration.

Numerical experiments across Bayesian regression, classification, and neural networks confirm these findings. The nonreversible samplers consistently show faster initial convergence compared to their reversible counterparts. In a quadratic model, the tuned skew perturbation reduces the number of iterations required to reach a target accuracy by several orders of magnitude compared to standard penalized Langevin dynamics.

This work provides a robust framework for constrained sampling that avoids the computational overhead of projections or proximal maps. By showing that nonreversible perturbations can be specifically designed to counteract the stiffness introduced by penalty methods, the paper offers a practical strategy for improving the efficiency of MCMC samplers in constrained parameter spaces. The results are particularly relevant for large-scale Bayesian learning where stochastic gradients are necessary.