Byzantine-Resilient Federated PCA and Low Rank Matrix Recovery

In this work we consider the problem of estimating the principal subspace (span of the top r singular vectors) of a symmetric matrix in a federated setting, when each node has access to estimates of this matrix. We study how to make this problem Byzantine resilient. We introduce a novel provably Byzantine-resilient, communication-efficient, and private algorithm, called Subspace-Median, to solve it. We also study the most natural solution for this problem, a geometric median based modification of the federated power method, and explain why it is not useful. We consider two special cases of the resilient subspace estimation meta-problem - federated principal components analysis (PCA) and the spectral initialization step of horizontally federated low rank column-wise sensing (LRCCS) in this work. For both these problems we show how Subspace Median provides a resilient solution that is also communication-efficient. Median of Means extensions are developed for both problems. Extensive simulation experiments are used to corroborate our theoretical guarantees. Our second contribution is a complete AltGDmin based algorithm for Byzantine-resilient horizontally federated LRCCS and guarantees for it. We do this by developing a geometric median of means estimator for aggregating the partial gradients computed at each node, and using Subspace Median for initialization.