  Singular value decomposition (SVD) is one of the most fundamental tools in
machine learning and statistics.The modern machine learning community usually
assumes that data come from and belong to small-scale device users. The low
communication and computation power of such devices, and the possible privacy
breaches of users' sensitive data make the computation of SVD challenging.
Federated learning (FL) is a paradigm enabling a large number of devices to
jointly learn a model in a communication-efficient way without data sharing. In
the FL framework, we develop a class of algorithms called FedPower for the
computation of partial SVD in the modern setting. Based on the well-known power
method, the local devices alternate between multiple local power iterations and
one global aggregation to improve communication efficiency. In the aggregation,
we propose to weight each local eigenvector matrix with Orthogonal Procrustes
Transformation (OPT). Considering the practical stragglers' effect, the
aggregation can be fully participated or partially participated, where for the
latter we propose two sampling and aggregation schemes. Further, to ensure
strong privacy protection, we add Gaussian noise whenever the communication
happens by adopting the notion of differential privacy (DP). We theoretically
show the convergence bound for FedPower. The resulting bound is interpretable
with each part corresponding to the effect of Gaussian noise, parallelization,
and random sampling of devices, respectively. We also conduct experiments to
demonstrate the merits of FedPower. In particular, the local iterations not
only improve communication efficiency but also reduce the chance of privacy
breaches.
