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：设计能够抵抗拜占庭敌手的分布式优化算法这一问题已受到广泛关注。在拜占庭容错分布式优化问题中，目标是（近似）最小化网络中常规（非对抗性）智能体所持有的局部成本函数的平均值。本文提出了一种通用的拜占庭容错分布式优化算法框架，该框架将一些最先进的算法作为特例。我们分析了框架内算法的收敛性，并推导出所有常规智能体以几何速率收敛到最优解附近一个球域（我们对其大小进行了刻画）。此外，我们证明在若干最小条件下可以几何快速地实现近似共识。我们的分析揭示了拜占庭容错分布式优化中收敛区域、常规智能体取值间距、步长以及智能体函数性质之间的关系。