  For engineering applications of artificial intelligence, Bayesian learning
holds significant advantages over standard frequentist learning, including the
capacity to quantify uncertainty. Langevin Monte Carlo (LMC) is an efficient
gradient-based approximate Bayesian learning strategy that aims at producing
samples drawn from the posterior distribution of the model parameters. Prior
work focused on a distributed implementation of LMC over a multi-access
wireless channel via analog modulation. In contrast, this paper proposes
quantized federated LMC (FLMC), which integrates one-bit stochastic
quantization of the local gradients with channel-driven sampling.
Channel-driven sampling leverages channel noise for the purpose of contributing
to Monte Carlo sampling, while also serving the role of privacy mechanism.
Analog and digital implementations of wireless LMC are compared as a function
of differential privacy (DP) requirements, revealing the advantages of the
latter at sufficiently high signal-to-noise ratio.
