Decentralized Proximal Stochastic Gradient Langevin Dynamics

Mohammad Rafiqul Islam; Lingjiong Zhu

stat.MLMay 1, 2026

Decentralized Proximal Stochastic Gradient Langevin Dynamics

Authors: Mohammad Rafiqul Islam, Lingjiong Zhu

Organizations: Department of Mathematics, Florida State University, Tallahassee, Florida, United States of America

Abstract

We propose Decentralized Proximal Stochastic Gradient Langevin Dynamics (DE-PSGLD), a decentralized Markov chain Monte Carlo (MCMC) algorithm for sampling from a log-concave probability distribution constrained to a convex domain. Constraints are enforced through a shared proximal regularization based on the Moreau-Yosida envelope, enabling unconstrained updates while preserving consistency with the target constrained posterior. We establish non-asymptotic convergence guarantees in the 2-Wasserstein distance for both individual agent iterates and their network averages. Our analysis shows that DE-PSGLD converges to a regularized Gibbs distribution and quantifies the bias introduced by the proximal approximation. We evaluate DE-PSGLD for different sampling problems on synthetic and real datasets. As the first decentralized approach for constrained domains, our algorithm exhibits fast posterior concentration and high predictive accuracy.

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Penalized Nonreversible Langevin for Constrained Sampling

Pervez Ali, Weihao Dong, Xiaoyu Wang

We propose penalized nonreversible Langevin algorithms for sampling from

π(x)\propto e^{-f(x)}\mathbf 1_{\mathcal C}(x)

, where

\mathcal C\subset\mathbb R^d

is a compact convex set. The algorithms combine a squared distance penalty with constant or compatible state dependent skew symmetric perturbations that preserve the penalized Gibbs distribution. For smooth, possibly nonconvex

f

, we derive nonasymptotic total variation bounds for the full gradient algorithm under a log Sobolev inequality. When unbiased stochastic gradients are available, we establish

2

-Wasserstein bounds under global contraction and Lipschitz conditions on the full drift in an adapted quadratic metric. For a fixed penalty parameter, the error relative to the penalized Gibbs distribution decays exponentially to an

\mathcal{O}(\sqrtη)

neighborhood, where

η

is the stepsize. We also bound the discrepancy between the penalized Gibbs distribution and the constrained target. In a two dimensional quadratic model, we establish nonreversible acceleration by tuning the skew perturbation to the curvature imbalance induced by penalization. With the target accuracy and smaller curvature fixed and initial Wasserstein distances uniformly bounded, tuning the skew perturbation improves the sufficient Euler iteration bound from linear to logarithmic in the curvature ratio. Numerical experiments evaluate the algorithms on constrained Bayesian regression, classification, neural networks, and truncated sampling, and examine the acceleration mechanism in a stochastic quadratic model.

Non-Log-Concave Proximal Gradient Descent

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Abstract

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