Approximate Byzantine Fault-Tolerance in Distributed Optimization

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 [24] 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 c 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; such as comparative gradient elimination or coordinate-wise trimmed mean.