DIGing--SGLD: Decentralized and Scalable Langevin Sampling over Time--Varying Networks

TL;DR

This paper introduces DIGing-SGLD, a decentralized Langevin sampling algorithm for Bayesian learning over time-varying networks, providing the first finite-time convergence guarantees and demonstrating strong empirical performance.

Contribution

It develops a novel decentralized SGLD algorithm that overcomes static network limitations and provides the first finite-time convergence analysis for time-varying networks.

Findings

Achieves geometric convergence to a neighborhood of the target distribution.

Provides explicit finite-time Wasserstein convergence guarantees.

Demonstrates strong empirical performance on Bayesian regression tasks.

Abstract

Sampling from a target distribution induced by training data is central to Bayesian learning, with Stochastic Gradient Langevin Dynamics (SGLD) serving as a key tool for scalable posterior sampling and decentralized variants enabling learning when data are distributed across a network of agents. This paper introduces DIGing-SGLD, a decentralized SGLD algorithm designed for scalable Bayesian learning in multi-agent systems operating over time-varying networks. Existing decentralized SGLD methods are restricted to static network topologies, and many exhibit steady-state sampling bias caused by network effects, even when full batches are used. DIGing-SGLD overcomes these limitations by integrating Langevin-based sampling with the gradient-tracking mechanism of the DIGing algorithm, originally developed for decentralized optimization over time-varying networks, thereby enabling efficient and bias-free sampling without a central coordinator. To our knowledge, we provide the first finite-time non-asymptotic Wasserstein convergence guarantees for decentralized SGLD-based sampling over time-varying networks, with explicit constants. Under standard strong convexity and smoothness assumptions, DIGing-SGLD achieves geometric convergence to an $O(\sqrt{\eta})$ neighborhood of the target distribution, where $\eta$ is the stepsize, with dependence on the target accuracy matching the best-known rates for centralized and static-network SGLD algorithms using constant stepsize. Numerical experiments on Bayesian linear and logistic regression validate the theoretical results and demonstrate the strong empirical performance of DIGing-SGLD under dynamically evolving network conditions.