  We study the problem of Byzantine fault tolerance in a distributed
optimization setting, where there is a group of $N$ agents communicating with a
trusted centralized coordinator. Among these agents, there is a subset of $f$
agents that may not follow a prescribed algorithm and may share arbitrarily
incorrect information with the coordinator. The goal is to find the optimizer
of the aggregate cost functions of the honest agents. We will be interested in
studying the local gradient descent method, also known as federated learning,
to solve this problem. However, this method often returns an approximate value
of the underlying optimal solution in the Byzantine setting. Recent work showed
that by incorporating the so-called comparative elimination (CE) filter at the
coordinator, one can provably mitigate the detrimental impact of Byzantine
agents and precisely compute the true optimizer in the convex setting. The
focus of the present work is to provide theoretical results to show the
convergence of local gradient methods with the CE filter in a nonconvex
setting. We will also provide a number of numerical simulations to support our
theoretical results.
