  Federated learning performed by a decentralized networks of agents is
becoming increasingly important with the prevalence of embedded software on
autonomous devices. Bayesian approaches to learning benefit from offering more
information as to the uncertainty of a random quantity, and Langevin and
Hamiltonian methods are effective at realizing sampling from an uncertain
distribution with large parameter dimensions. Such methods have only recently
appeared in the decentralized setting, and either exclusively use stochastic
gradient Langevin and Hamiltonian Monte Carlo approaches that require a
diminishing stepsize to asymptotically sample from the posterior and are known
in practice to characterize uncertainty less faithfully than constant step-size
methods with a Metropolis adjustment, or assume strong convexity properties of
the potential function. We present the first approach to incorporating constant
stepsize Metropolis-adjusted HMC in the decentralized sampling framework, show
theoretical guarantees for consensus and probability distance to the posterior
stationary distribution, and demonstrate their effectiveness numerically on
standard real world problems, including decentralized learning of neural
networks which is known to be highly non-convex.
