The problem of distributed optimization requires a group of agents to reach agreement on a parameter that minimizes the average of their local cost functions using information received from their neighbors. While there are a variety of distributed optimization algorithms that can solve this problem, they are typically vulnerable to malicious (or ``Byzantine'') agents that do not follow the algorithm. Recent attempts to address this issue focus on single dimensional functions, or provide analysis under certain assumptions on the statistical properties of the functions at the agents. In this paper, we propose a resilient distributed optimization algorithm for multi-dimensional convex functions. Our scheme involves two filtering steps at each iteration of the algorithm: (1) distance-based and (2) component-wise removal of extreme states. We show that this algorithm can mitigate the impact of up to $F$ Byzantine agents in the neighborhood of each regular node, without knowing the identities of the Byzantine agents in advance. In particular, we show that if the network topology satisfies certain conditions, all of the regular states are guaranteed to asymptotically converge to a bounded region that contains the global minimizer.

翻译：分布式优化问题要求一组智能体利用从邻居接收的信息，就最小化其局部代价函数平均值的参数达成一致。尽管存在多种可解决该问题的分布式优化算法，但这些算法通常易受不遵循算法的恶意（或“拜占庭”）智能体攻击。近期解决该问题的尝试主要聚焦于单维函数，或对智能体函数的统计特性作出特定假设。本文针对多维凸函数提出了一种弹性分布式优化算法。我们的方案在算法每次迭代中包括两个过滤步骤：（1）基于距离的极端状态剔除，及（2）逐分量极端状态剔除。我们证明，该算法可在每个常规节点邻域内抵御多达$F$个拜占庭智能体的影响，且无需预先知晓拜占庭智能体的身份。特别地，若网络拓扑满足特定条件，所有常规状态保证渐近收敛至包含全局最小化器的有界区域。