DIGing--SGLD: Decentralized and Scalable Langevin Sampling over Time--Varying Networks
This work addresses the need for bias-free and efficient sampling in decentralized Bayesian learning for multi-agent systems, overcoming limitations of static networks and network-induced bias, though it is incremental as it builds on existing SGLD and DIGing methods.
The paper tackles the problem of scalable Bayesian learning in multi-agent systems over time-varying networks by introducing DIGing-SGLD, a decentralized algorithm that integrates Langevin sampling with gradient-tracking, achieving geometric convergence to an O(√η) neighborhood of the target distribution with explicit finite-time guarantees.
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 $O(\sqrtη)$ neighborhood of the target distribution, where $η$ 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.