  We consider the problem of Byzantine fault-tolerance in federated machine
learning. In this problem, the system comprises multiple agents each with local
data, and a trusted centralized coordinator. In fault-free setting, the agents
collaborate with the coordinator to find a minimizer of the aggregate of their
local cost functions defined over their local data. We consider a scenario
where some agents ($f$ out of $N$) are Byzantine faulty. Such agents need not
follow a prescribed algorithm correctly, and may communicate arbitrary
incorrect information to the coordinator. In the presence of Byzantine agents,
a more reasonable goal for the non-faulty agents is to find a minimizer of the
aggregate cost function of only the non-faulty agents. This particular goal is
commonly referred as exact fault-tolerance. Recent work has shown that exact
fault-tolerance is achievable if only if the non-faulty agents satisfy the
property of $2f$-redundancy. Now, under this property, techniques are known to
impart exact fault-tolerance to the distributed implementation of the classical
stochastic gradient-descent (SGD) algorithm. However, we do not know of any
such techniques for the federated local SGD algorithm - a more commonly used
method for federated machine learning. To address this issue, we propose a
novel technique named comparative elimination (CE). We show that, under
$2f$-redundancy, the federated local SGD algorithm with CE can indeed obtain
exact fault-tolerance in the deterministic setting when the non-faulty agents
can accurately compute gradients of their local cost functions. In the general
stochastic case, when agents can only compute unbiased noisy estimates of their
local gradients, our algorithm achieves approximate fault-tolerance with
approximation error proportional to the variance of stochastic gradients and
the fraction of Byzantine agents.
