Fast and privacy preserving distributed low-rank regression

This paper proposes a fast and privacy preserving distributed algorithm for handling low-rank regression problems with nuclear norm constraint. Traditional projected gradient algorithms have high computation costs due to their projection steps when they are used to solve these problems. Our gossip-based algorithm, called the fast DeFW algorithm, overcomes this issue since it is projection-free. In particular, the algorithm incorporates a carefully designed decentralized power method step to reduce the complexity by distributed computation over network. Meanwhile, privacy is preserved as the agents do not exchange the private data, but only a random projection of them. We show that the fast DeFW algorithm converges for both convex and non-convex losses. As an application example, we consider the low-rank matrix completion problem and provide numerical results to support our findings.