This paper proposes a Robust One-Step Estimator(ROSE) to solve the Byzantine failure problem in distributed M-estimation when a moderate fraction of node machines experience Byzantine failures. To define ROSE, the algorithms use the robust Variance Reduced Median Of the Local(VRMOL) estimator to determine the initial parameter value for iteration, and communicate between the node machines and the central processor in the Newton-Raphson iteration procedure to derive the robust VRMOL estimator of the gradient, and the Hessian matrix so as to obtain the final estimator. ROSE has higher asymptotic relative efficiency than general median estimators without increasing the order of computational complexity. Moreover, this estimator can also cope with the problems involving anomalous or missing samples on the central processor. We prove the asymptotic normality when the parameter dimension p diverges as the sample size goes to infinity, and under weaker assumptions, derive the convergence rate. Numerical simulations and a real data application are conducted to evidence the effectiveness and robustness of ROSE.

翻译：本文提出一种鲁棒一步估计器（ROSE），用于解决当部分节点机器发生中等级别拜占庭故障时分布式M估计中的拜占庭故障问题。为定义ROSE，算法采用鲁棒方差缩减局部中位数（VRMOL）估计器确定迭代初始参数值，并在牛顿-拉夫逊迭代过程中通过节点机器与中央处理器之间的通信，推导出梯度的鲁棒VRMOL估计量及海森矩阵，从而获得最终估计量。与普通中位数估计器相比，ROSE在不增加计算复杂度阶数的前提下具有更高的渐近相对效率。此外，该估计器还能处理中央处理器上出现的异常或缺失样本问题。我们证明了当参数维度p随样本量趋于无穷大而发散时的渐近正态性，并在更弱假设下推导了收敛速率。通过数值模拟和实际数据应用验证了ROSE的有效性与鲁棒性。