  We propose a federated averaging Langevin algorithm (FA-LD) for uncertainty
quantification and mean predictions with distributed clients. In particular, we
generalize beyond normal posterior distributions and consider a general class
of models. We develop theoretical guarantees for FA-LD for strongly log-concave
distributions with non-i.i.d data and study how the injected noise and the
stochastic-gradient noise, the heterogeneity of data, and the varying learning
rates affect the convergence. Such an analysis sheds light on the optimal
choice of local updates to minimize communication costs. Important to our
approach is that the communication efficiency does not deteriorate with the
injected noise in the Langevin algorithms. In addition, we examine in our FA-LD
algorithm both independent and correlated noise used over different clients. We
observe there is a trade-off between the pairs among communication, accuracy,
and data privacy. As local devices may become inactive in federated networks,
we also show convergence results based on different averaging schemes where
only partial device updates are available. In such a case, we discover an
additional bias that does not decay to zero.
