This paper considers the problem of Byzantine fault-tolerance in distributed multi-agent optimization. In this problem, each agent has a local cost function, and in the fault-free case, the goal is to design a distributed algorithm that allows all the agents to find a minimum point of all the agents aggregate cost function. We consider a scenario where some agents might be Byzantine faulty that renders the original goal of computing a minimum point of all the agents aggregate cost vacuous. A more reasonable objective for an algorithm in this scenario is to allow all the non-faulty agents to compute the minimum point of only the non-faulty agents aggregate cost. Prior work shows that if there are up to $f$ (out of $n$) Byzantine agents then a minimum point of the non-faulty agents aggregate cost can be computed exactly if and only if the non-faulty agents costs satisfy a certain redundancy property called $2f$-redundancy. However, $2f$-redundancy is an ideal property that can be satisfied only in systems free from noise or uncertainties, which can make the goal of exact fault-tolerance unachievable in some applications. Thus, we introduce the notion of $(f,epsilon)$-resilience, a generalization of exact fault-tolerance wherein the objective is to find an approximate minimum point of the non-faulty aggregate cost, with $epsilon$ accuracy. This approximate fault-tolerance can be achieved under a weaker condition that is easier to satisfy in practice, compared to $2f$-redundancy. We obtain necessary and sufficient conditions for achieving $(f,epsilon)$-resilience characterizing the correlation between relaxation in redundancy and approximation in resilience. In case when the agents cost functions are differentiable, we obtain conditions for $(f,epsilon)$-resilience of the distributed gradient-descent method when equipped with robust gradient aggregation.
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