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Most works on federated learning (FL) focus on the most common frequentistformulation of learning whereby the goal is minimizing the global empiricalloss. Frequentist learning, however, is known to be problematic in the regimeof limited data as it fails to quantify epistemic uncertainty in prediction.Bayesian learning provides a principled solution to this problem by shiftingthe optimization domain to the space of distribution in the model parameters.{\color{black}This paper proposes a novel mechanism for the efficientimplementation of Bayesian learning in wireless systems. Specifically, we focuson a standard gradient-based Markov Chain Monte Carlo (MCMC) method, namelyLangevin Monte Carlo (LMC), and we introduce a novel protocol, termed WirelessFederated LMC (WFLMC), that is able to repurpose channel noise for the doublerole of seed randomness for MCMC sampling and of privacy preservation.} To thisend, based on the analysis of the Wasserstein distance between sampledistribution and global posterior distribution under privacy and powerconstraints, we introduce a power allocation strategy as the solution of aconvex program. The analysis identifies distinct operating regimes in which theperformance of the system is power-limited, privacy-limited, or limited by therequirement of MCMC sampling. Both analytical and simulation resultsdemonstrate that, if the channel noise is properly accounted for under suitableconditions, it can be fully repurposed for both MCMC sampling and privacypreservation, obtaining the same performance as in an ideal communicationsetting that is not subject to privacy constraints.

Wireless Federated Langevin Monte Carlo: Repurposing Channel Noise for Bayesian Sampling and Privacy