Bayesian Federated Learning (FL) has been recently introduced to provide well-calibrated Machine Learning (ML) models quantifying the uncertainty of their predictions. Despite their advantages compared to frequentist FL setups, Bayesian FL tools implemented over decentralized networks are subject to high communication costs due to the iterated exchange of local posterior distributions among cooperating devices. Therefore, this paper proposes a communication-efficient decentralized Bayesian FL policy to reduce the communication overhead without sacrificing final learning accuracy and calibration. The proposed method integrates compression policies and allows devices to perform multiple optimization steps before sending the local posterior distributions. We integrate the developed tool in an Industrial Internet of Things (IIoT) use case where collaborating nodes equipped with autonomous radar sensors are tasked to reliably localize human operators in a workplace shared with robots. Numerical results show that the developed approach obtains highly accurate yet well-calibrated ML models compatible with the ones provided by conventional (uncompressed) Bayesian FL tools while substantially decreasing the communication overhead (i.e., up to 99%). Furthermore, the proposed approach is advantageous when compared with state-of-the-art compressed frequentist FL setups in terms of calibration, especially when the statistical distribution of the testing dataset changes.

Stochastic gradient MCMC methods, such as stochastic gradient Langevin dynamics (SGLD), enable large-scale posterior inference by leveraging noisy but cheap gradient estimates. However, when federated data are non-IID, the variance of distributed gradient estimates is amplified compared to its centralized version, and delayed communication rounds lead chains to diverge from the target posterior. In this work, we introduce the concept of conducive gradients, zero-mean stochastic gradients that serve as a mechanism for sharing probabilistic information between data shards. We propose a novel stochastic gradient estimator that incorporates the conducive gradients, and we show that it improves convergence on federated data when compared to distributed SGLD (DSGLD). We evaluate, conducive gradient DSGLD (CG-DSGLD) on metric learning and deep MLPs tasks. Experiments show that it outperforms standard DSGLD for non-IID federated data.