  Learning a privacy-preserving model from sensitive data which are distributed
across multiple devices is an increasingly important problem. The problem is
often formulated in the federated learning context, with the aim of learning a
single global model while keeping the data distributed. Moreover, Bayesian
learning is a popular approach for modelling, since it naturally supports
reliable uncertainty estimates. However, Bayesian learning is generally
intractable even with centralised non-private data and so approximation
techniques such as variational inference are a necessity. Variational inference
has recently been extended to the non-private federated learning setting via
the partitioned variational inference algorithm. For privacy protection, the
current gold standard is called differential privacy. Differential privacy
guarantees privacy in a strong, mathematically clearly defined sense.
  In this paper, we present differentially private partitioned variational
inference, the first general framework for learning a variational approximation
to a Bayesian posterior distribution in the federated learning setting while
minimising the number of communication rounds and providing differential
privacy guarantees for data subjects.
  We propose three alternative implementations in the general framework, one
based on perturbing local optimisation runs done by individual parties, and two
based on perturbing updates to the global model (one using a version of
federated averaging, the second one adding virtual parties to the protocol),
and compare their properties both theoretically and empirically.
