In this paper, we consider a min-max optimization problem under adversarial manipulation, where there are $n$ cost functions, up to $f$ of which may be replaced by arbitrary faulty functions by an adversary. The goal is to minimize the maximum cost over $x$ among the $n$ functions despite the faulty functions. The problem formulation could naturally extend to Byzantine fault-tolerant distributed min-max optimization. We present a simple algorithm for Byzantine min-max optimization, and provide bounds on the output of the algorithm. We also present an approximate algorithm for this problem. We then extend the problem to a distributed setting and present a distributed algorithm. To the best of our knowledge, we are the first to consider this problem.

翻译：本文研究了对抗性操纵下的极小极大（min-max）优化问题，其中共有$n$个代价函数，攻击者可能将其中至多$f$个函数替换为任意故障函数。目标是在存在故障函数的情况下，最小化$n$个函数中关于$x$的最大代价。该问题形式可自然扩展至拜占庭容错分布式极小极大优化。我们提出了一种用于拜占庭极小极大优化的简单算法，并给出了算法输出的界值。同时，针对该问题提出了一种近似算法。随后，我们将该问题扩展至分布式场景，并设计了一种分布式算法。据我们所知，这是首次对该问题进行研究。