Decentralized Proximal Stochastic Gradient Langevin Dynamics

TL;DR

This paper introduces DE-PSGLD, a decentralized MCMC algorithm for sampling from constrained log-concave distributions, with proven convergence guarantees and demonstrated effectiveness on synthetic and real data.

Contribution

It presents the first decentralized algorithm for constrained domains, combining proximal regularization with stochastic gradient Langevin dynamics, and provides theoretical convergence analysis.

Findings

DE-PSGLD converges to a regularized Gibbs distribution.

The algorithm exhibits fast posterior concentration.

It achieves high predictive accuracy on various datasets.

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.