The problem of designing distributed optimization algorithms that are resilient to Byzantine adversaries has received significant attention. For the Byzantine-resilient distributed optimization problem, the goal is to (approximately) minimize the average of the local cost functions held by the regular (non adversarial) agents in the network. In this paper, we provide a general algorithmic framework for Byzantine-resilient distributed optimization which includes some state-of-the-art algorithms as special cases. We analyze the convergence of algorithms within the framework, and derive a geometric rate of convergence of all regular agents to a ball around the optimal solution (whose size we characterize). Furthermore, we show that approximate consensus can be achieved geometrically fast under some minimal conditions. Our analysis provides insights into the relationship among the convergence region, distance between regular agents' values, step-size, and properties of the agents' functions for Byzantine-resilient distributed optimization.

翻译：设计对拜占庭攻击具有鲁棒性的分布式优化算法问题已受到广泛关注。对于拜占庭鲁棒分布式优化问题，其目标是（近似）最小化网络中正常（非恶意）智能体局部成本函数的平均值。本文提出了一种通用的拜占庭鲁棒分布式优化算法框架，该框架将若干当前最优算法作为特例纳入其中。我们分析了框架内算法的收敛性，并推导出所有正常智能体以几何速率收敛到最优解附近一个球域（其规模已刻画）的结论。此外，我们证明了在若干最小条件下可快速实现近似一致性。我们的分析揭示了拜占庭鲁棒分布式优化中收敛区域、正常智能体数值间距、步长以及智能体函数性质之间的关联。