Decentralized Proximal Stochastic Gradient Langevin Dynamics
Mohammad Rafiqul Islam, Lingjiong Zhu
TLDR
DE-PSGLD is a novel decentralized MCMC algorithm for sampling from constrained log-concave distributions, offering strong convergence guarantees.
Key contributions
- Proposes DE-PSGLD, a decentralized MCMC for sampling constrained log-concave distributions.
- Enforces constraints using shared proximal regularization via the Moreau-Yosida envelope.
- Establishes non-asymptotic 2-Wasserstein convergence for agent iterates and network averages.
- Quantifies the bias introduced by the proximal approximation, converging to a regularized Gibbs distribution.
Why it matters
This paper introduces the first decentralized MCMC algorithm capable of handling constrained sampling problems, a significant advancement for distributed inference. By using proximal regularization, it enables efficient updates while maintaining consistency with target posteriors. This leads to faster posterior concentration and improved predictive accuracy in practical applications.
Original 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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