  Federated learning (FL) allows multiple clients to collaboratively learn a
globally shared model through cycles of model aggregation and local model
training without the need to share data. In this paper, we comprehensively
study a new problem named aggregation error (AE), arising from the model
aggregation stage on a server, which is mainly induced by the heterogeneity of
the client data. Due to the large discrepancies between local models, the
accompanying large AE generally results in a slow convergence and an expected
reduction of accuracy for FL. In order to reduce AE, we propose a novel
federated learning framework from a Bayesian perspective, in which a
multivariate Gaussian product mechanism is employed to aggregate the local
models. It is worth noting that the product of Gaussians is still a Gaussian.
This property allows us to directly aggregate local expectations and
covariances in a definitely convex form, thereby greatly reducing the AE.
Accordingly, on the clients, we develop a new Federated Online Laplace
Approximation (FOLA) method, which can estimate the parameters of the local
posterior by repeatedly accumulating priors. Specifically, in every round, the
global posterior distributed from the server can be treated as the priors, and
thus the local posterior can also be effectively approximated by a Gaussian
using FOLA. Experimental results on benchmarks reach state-of-the-arts
performance and clearly demonstrate the advantages of the proposed method.
