  Both Byzantine resilience and communication efficiency have attracted
tremendous attention recently for their significance in edge federated
learning. However, most existing algorithms may fail when dealing with
real-world irregular data that behaves in a heavy-tailed manner. To address
this issue, we study the stochastic convex and non-convex optimization problem
for federated learning at edge and show how to handle heavy-tailed data while
retaining the Byzantine resilience, communication efficiency and the optimal
statistical error rates simultaneously. Specifically, we first present a
Byzantine-resilient distributed gradient descent algorithm that can handle the
heavy-tailed data and meanwhile converge under the standard assumptions. To
reduce the communication overhead, we further propose another algorithm that
incorporates gradient compression techniques to save communication costs during
the learning process. Theoretical analysis shows that our algorithms achieve
order-optimal statistical error rate in presence of Byzantine devices. Finally,
we conduct extensive experiments on both synthetic and real-world datasets to
verify the efficacy of our algorithms.
