  We consider a distributed learning problem, where agents minimize a global
objective function by exchanging information over a network. Our approach has
two distinct features: (i) It substantially reduces communication by triggering
communication only when necessary, and (ii) it is agnostic to the
data-distribution among the different agents. We can therefore guarantee
convergence even if the local data-distributions of the agents are arbitrarily
distinct. We analyze the convergence rate of the algorithm and derive
accelerated convergence rates in a convex setting. We also characterize the
effect of communication drops and demonstrate that our algorithm is robust to
communication failures. The article concludes by presenting numerical results
from a distributed LASSO problem, and distributed learning tasks on MNIST and
CIFAR-10 datasets. The experiments underline communication savings of 50% or
more due to the event-based communication strategy, show resilience towards
heterogeneous data-distributions, and highlight that our approach outperforms
common baselines such as FedAvg, FedProx, and FedADMM.
