  We study the problem of distributed cooperative learning, where a group of
agents seek to agree on a set of hypotheses that best describes a sequence of
private observations. In the scenario where the set of hypotheses is large, we
propose a belief update rule where agents share compressed (either sparse or
quantized) beliefs with an arbitrary positive compression rate. Our algorithm
leverages a unified and straightforward communication rule that enables agents
to access wide-ranging compression operators as black-box modules. We prove the
almost sure asymptotic exponential convergence of beliefs around the set of
optimal hypotheses. Additionally, we show a non-asymptotic, explicit, and
linear concentration rate in probability of the beliefs on the optimal
hypothesis set. We provide numerical experiments to illustrate the
communication benefits of our method. The simulation results show that the
number of transmitted bits can be reduced to 5-10% of the non-compressed method
in the studied scenarios.
