This paper addresses the problem of robust estimation in gossip algorithms over arbitrary communication graphs. Gossip algorithms are fully decentralized, relying only on local neighbor-to-neighbor communication, making them well-suited for situations where communication is constrained. A fundamental challenge in existing mean-based gossip algorithms is their vulnerability to malicious or corrupted nodes. In this paper, we show that an outlier-robust mean can be computed by globally estimating a robust statistic. More specifically, we propose a novel gossip algorithm for rank estimation, referred to as \textsc{GoRank}, and leverage it to design a gossip procedure dedicated to trimmed mean estimation, coined \textsc{GoTrim}. In addition to a detailed description of the proposed methods, a key contribution of our work is a precise convergence analysis: we establish an $\mathcal{O}(1/t)$ rate for rank estimation and an $\mathcal{O}(1 / {t})$ rate for trimmed mean estimation, where by $t$ is meant the number of iterations. Moreover, we provide a breakdown point analysis of \textsc{GoTrim}. We empirically validate our theoretical results through experiments on diverse network topologies, data distributions and contamination schemes.

翻译：本文研究了任意通信图上的Gossip算法中的鲁棒估计问题。Gossip算法完全去中心化，仅依赖局部邻居节点间的通信，因此特别适用于通信受限的场景。现有基于均值的Gossip算法面临的一个根本性挑战在于其对恶意或损坏节点的脆弱性。本文证明，通过全局估计鲁棒统计量可以计算出抗异常值的均值。具体而言，我们提出了一种用于秩估计的新型Gossip算法（称为\textsc{GoRank}），并利用该算法设计了专用于截尾均值估计的Gossip流程（命名为\textsc{GoTrim}）。除了对所提方法的详细描述外，本研究的关键贡献在于精确的收敛性分析：我们确立了秩估计的$\mathcal{O}(1/t)$收敛速率与截尾均值估计的$\mathcal{O}(1 / {t})$收敛速率，其中$t$表示迭代次数。此外，我们还提供了\textsc{GoTrim}的崩溃点分析。通过在多样化网络拓扑、数据分布和污染方案上的实验，我们对理论结果进行了实证验证。