Decentralized Proximal Stochastic Gradient Langevin Dynamics

2026-05-01

AI Summary

This work addresses the challenge of efficiently sampling from log-concave distributions over convex constrained domains in decentralized settings. It proposes the first decentralized Markov Chain Monte Carlo (MCMC) algorithm tailored for constrained domains, leveraging a shared Moreau–Yosida proximal regularization to transform the constrained problem into unconstrained local updates. This approach enables collaborative sampling across nodes while preserving consistency with the target posterior distribution. Theoretical analysis establishes non-asymptotic convergence guarantees and explicitly quantifies the bias introduced by the proximal approximation. Empirical evaluations on both synthetic and real-world datasets demonstrate that the algorithm rapidly converges to the true posterior and achieves high predictive accuracy.

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.

Problem

• decentralized sampling
• constrained posterior
• log-concave distribution
• convex domain
• Markov chain Monte Carlo

Innovation

• Decentralized MCMC
• Proximal regularization
• Stochastic Gradient Langevin Dynamics
• Constrained sampling
• Moreau-Yosida envelope