This work considers two related learning problems in a federated attack prone setting: federated principal components analysis (PCA) and federated low rank column-wise sensing (LRCS). The node attacks are assumed to be Byzantine which means that the attackers are omniscient and can collude. We introduce a novel provably Byzantine-resilient communication-efficient and sampleefficient algorithm, called Subspace-Median, that solves the PCA problem and is a key part of the solution for the LRCS problem. We also study the most natural Byzantine-resilient solution for federated PCA, a geometric median based modification of the federated power method, and explain why it is not useful. Our second main contribution is a complete alternating gradient descent (GD) and minimization (altGDmin) algorithm for Byzantine-resilient horizontally federated LRCS and sample and communication complexity guarantees for it. Extensive simulation experiments are used to corroborate our theoretical guarantees. The ideas that we develop for LRCS are easily extendable to other LR recovery problems as well.

翻译：本研究探讨了在易受攻击的联邦学习环境下的两个相关学习问题：联邦主成分分析（PCA）与联邦低秩列感知（LRCS）。节点攻击被假定为拜占庭式，这意味着攻击者具备全知能力且可共谋。我们提出了一种新颖的、可证明具有拜占庭鲁棒性、通信高效且样本高效的算法——子空间中位数法（Subspace-Median），该算法解决了PCA问题，并成为LRCS解决方案的关键组成部分。同时，我们研究了联邦PCA最自然的拜占庭鲁棒解决方案——基于几何中位数的联邦幂方法改进，并阐释了其局限性。我们的第二个主要贡献是提出了完整的交替梯度下降（GD）与最小化（altGDmin）算法，用于拜占庭鲁棒的水平联邦LRCS，并给出了其样本与通信复杂度保证。大量仿真实验验证了我们的理论保证。我们为LRCS开发的思想可轻松扩展至其他低秩恢复问题。