This report considers the problem of Byzantine fault-tolerance in multi-agent collaborative optimization. In this problem, each agent has a local cost function. The goal of a collaborative optimization algorithm is to compute a minimum of the aggregate of the agents' cost functions. We consider the case when a certain number of agents may be Byzantine faulty. Such faulty agents may not follow a prescribed algorithm, and they may send arbitrary or incorrect information regarding their local cost functions. A reasonable goal in presence of such faulty agents is to minimize the aggregate cost of the non-faulty agents. In this report, we show that this goal can be achieved if and only if the cost functions of the non-faulty agents have a minimal redundancy property. We present different algorithms that achieve such tolerance against faulty agents, and demonstrate a trade-off between the complexity of an algorithm and the properties of the agents' cost functions. Further, we also consider the case when the cost functions are independent or do not satisfy the minimal redundancy property. In that case, we quantify the tolerance against faulty agents by introducing a metric called weak resilience. We present an algorithm that attains weak resilience when the faulty agents are in the minority and the cost functions are non-negative.

中文翻译：

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协同优化的弹性：冗余和独立的成本函数

本报告考虑了多智能体协同优化中的拜占庭容错问题。在这个问题中，每个代理都有一个本地成本函数。协同优化算法的目标是计算代理成本函数的最小值。我们考虑一定数量的代理可能是拜占庭故障的情况。此类有缺陷的代理可能不遵循规定的算法，并且他们可能会发送有关其本地成本函数的任意或不正确的信息。存在此类故障代理的合理目标是最小化非故障代理的总成本。在本报告中，我们表明当且仅当非故障代理的成本函数具有最小冗余属性时，才能实现该目标。我们提出了不同的算法来实现对错误代理的容忍度，并证明算法的复杂性与代理成本函数的属性之间的权衡。此外，我们还考虑了成本函数独立或不满足最小冗余属性的情况。在这种情况下，我们通过引入称为弱弹性的指标来量化对故障代理的容忍度。我们提出了一种算法，当故障代理为少数且成本函数为非负时，该算法获得弱弹性。