  Compression schemes have been extensively used in Federated Learning (FL) to
reduce the communication cost of distributed learning. While most approaches
rely on a bounded variance assumption of the noise produced by the compressor,
this paper investigates the use of compression and aggregation schemes that
produce a specific error distribution, e.g., Gaussian or Laplace, on the
aggregated data. We present and analyze different aggregation schemes based on
layered quantizers achieving exact error distribution. We provide different
methods to leverage the proposed compression schemes to obtain
compression-for-free in differential privacy applications. Our general
compression methods can recover and improve standard FL schemes with Gaussian
perturbations such as Langevin dynamics and randomized smoothing.
