  Federated sampling algorithms have recently gained great popularity in the
community of machine learning and statistics. This paper studies variants of
such algorithms called Error Feedback Langevin algorithms (ELF). In particular,
we analyze the combinations of EF21 and EF21-P with the federated Langevin
Monte-Carlo. We propose three algorithms: P-ELF, D-ELF, and B-ELF that use,
respectively, primal, dual, and bidirectional compressors. We analyze the
proposed methods under Log-Sobolev inequality and provide non-asymptotic
convergence guarantees.
