Federated PCA with Adaptive Rank Estimation

In many online machine learning and data science tasks such as data summarisation and feature compression, $d$-dimensional vectors are usually distributed across a large number of clients in a decentralised network and collected in a streaming fashion. This is increasingly common in modern applications due to the sheer volume of data generated and the clients' constrained resources. In this setting, some clients are required to compute an update to a centralised target model independently using local data while other clients aggregate these updates with a low-complexity merging algorithm. However, some clients with limited storage might not be able to store all of the data samples if $d$ is large, nor compute procedures requiring at least $\Omega(d^2)$ storage-complexity such as Principal Component Analysis, Subspace Tracking, or general Feature Correlation. In this work, we present a novel federated algorithm for PCA that is able to adaptively estimate the rank $r$ of the dataset and compute its $r$ leading principal components when only $O(dr)$ memory is available. This inherent adaptability implies that $r$ does not have to be supplied as a fixed hyper-parameter which is beneficial when the underlying data distribution is not known in advance, such as in a streaming setting. Numerical simulations show that, while using limited-memory, our algorithm exhibits state-of-the-art performance that closely matches or outperforms traditional non-federated algorithms, and in the absence of communication latency, it exhibits attractive horizontal scalability.