Approximate Byzantine Fault-Tolerance in Distributed Optimization

We consider 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 up to $f$ (out of $n$) agents might be Byzantine faulty, i.e., these agents may not follow a prescribed algorithm and may share arbitrary information regarding their local cost functions. In the presence of such faulty agents, a more reasonable goal is to design an algorithm that allows all the non-faulty agents to compute, either exactly or approximately, the minimum point of only the non-faulty agents' aggregate cost function. From recent work we know that a deterministic algorithm can compute a minimum point of the non-faulty agents' aggregate cost exactly if and only if the non-faulty agents' cost functions satisfy a certain redundancy property named $2f$-redundancy. However, the $2f$-redundancy property can only be guaranteed in ideal systems free from noises, and thus, exact fault-tolerance is unsuitable for many practical settings. In this paper, we consider the problem of approximate fault-tolerance - a generalization of exact fault-tolerance where the goal is to only compute an approximation of a minimum point. We define approximate fault-tolerance formally as $(f, \, \epsilon)$-resilience where $\epsilon$ is the approximation error, and we show that it can be achieved under a weaker redundancy condition than $2f$-redundancy. In the special case when the cost functions are differentiable, we analyze the approximate fault-tolerance of the distributed gradient-descent method equipped with a gradient-filter; such as comparative gradient elimination (CGE) or coordinate-wise trimmed mean (CWTM).