We study distributed principal component analysis (PCA) for large-scale federated data when the sample size $n$n and dimension $d$d are both ultra-large. This type of data is currently very common, but faces numerous challenges in PCA learning, such as communication overhead and computational complexity. We develop a new algorithm ${\mathsf {FedFask}}$FedFask (Fast Sketching for Federated learning) with lower communication cost $O(dr)$O(dr) and lower computational complexity $O(d(np/m+p^{2}+r^{2}))$O(d(np/m+p2+r2)), where $m$m is the number of workers, $r$r is the rank of matrix, $p$p is the dimension of sketched column space, and $r\leq p\ll d$r≤p≪d. In ${\mathsf {FedFask}}$FedFask, we adopt and develop technologies such as fast sketching, alignments with orthogonal Procrustes Fixing, and matrix Stiefel manifold via Kolmogorov-Nagumo-type average. Thus, ${\mathsf {FedFask}}$FedFask has a higher accuracy, lower stochastic variation, and best representation of multiple randomly projected eigenspaces, and avoids the orthogonal ambiguity of eigenspaces. We show that ${\mathsf {FedFask}}$FedFask achieves the same rate of learning $O\left(\frac{\kappa _{r}r}{\lambda _{r}}\sqrt{\frac{r^{*}}{n}}\right)$Oκrrλrr*n as the centralized PCA uses all data, and tolerates more workers to parallel acceleration computation. We conduct extensive experiments to demonstrate the effectiveness of ${\mathsf {FedFask}}$FedFask.

Facing large-scale online learning, the reliance on sophisticated model architectures often leads to nonconvex distributed optimization, which is more challenging than convex problems. Online recruited workers, such as mobile phone, laptop, and desktop computers, often have narrower uplink bandwidths than downlink. In this article, we propose two communication-efficient nonconvex federated learning algorithms with error feedback 2021 (EF21) and lazily aggregated gradient (LAG) for adapting uplink and downlink communications. EF21 is a new and theoretically better EF, which consistently and substantially outperforms vanilla EF in practice. LAG is a gradient filtration technique for adapting communication. For reducing communication costs of uplink, we design an effective LAG rule and then give EF21 with LAG (EF-LAG) algorithm, which combines EF21 and our LAG rule. We also present a bidirectional EF-LAG (BiEF-LAG) algorithm for reducing uplink and downlink communication costs. Theoretically, our proposed algorithms enjoy the same fast convergence rate as gradient descent (GD) for smooth nonconvex learning. That is, our algorithms greatly reduce communication costs without sacrificing the quality of learning. Numerical experiments on both synthetic data and deep learning benchmarks show significant empirical superiority of our algorithms in communication.