Byzantine Fault-Tolerance in Federated Local SGD Under $2f$-Redundancy

In this article, we study the problem of Byzantine fault-tolerance in a federated optimization setting, where there is a group of agents communicating with a centralized coordinator. We allow up to
$f$
Byzantine-faulty agents, which may not follow a prescribed algorithm correctly, and may share arbitrarily incorrect information with the coordinator. Associated with each nonfaulty agent is a local cost function. The goal of the nonfaulty agents is to compute a minimizer of their aggregate cost function. For solving this problem, we propose a local gradient-descent algorithm that incorporates a novel
comparative elimination
filter (aka. aggregation scheme) to provably mitigate the detrimental impact of Byzantine faults. In the deterministic setting, when the agents can compute their local gradients accurately, our algorithm guarantees
exact
fault-tolerance against a bounded fraction of Byzantine agents, provided the nonfaulty agents satisfy the known necessary condition of
$2f$-redundancy
. In the stochastic setting, when the agents can only compute stochastic estimates of their gradients, our algorithm guarantees
approximate
fault-tolerance where the approximation error is proportional to the variance of stochastic gradients and the fraction of Byzantine agents.