  Stochastic gradient Langevin dynamics (SGLD) and stochastic gradient
Hamiltonian Monte Carlo (SGHMC) are two popular Markov Chain Monte Carlo (MCMC)
algorithms for Bayesian inference that can scale to large datasets, allowing to
sample from the posterior distribution of the parameters of a statistical model
given the input data and the prior distribution over the model parameters.
However, these algorithms do not apply to the decentralized learning setting,
when a network of agents are working collaboratively to learn the parameters of
a statistical model without sharing their individual data due to privacy
reasons or communication constraints. We study two algorithms: Decentralized
SGLD (DE-SGLD) and Decentralized SGHMC (DE-SGHMC) which are adaptations of SGLD
and SGHMC methods that allow scaleable Bayesian inference in the decentralized
setting for large datasets. We show that when the posterior distribution is
strongly log-concave and smooth, the iterates of these algorithms converge
linearly to a neighborhood of the target distribution in the 2-Wasserstein
distance if their parameters are selected appropriately. We illustrate the
efficiency of our algorithms on decentralized Bayesian linear regression and
Bayesian logistic regression problems.
