Fast and Robust Low-Rank Learning over Networks: A Decentralized Matrix Quantile Regression Approach

Decentralized low-rank learning is an active research domain with extensive practical applications. A common approach to producing low-rank and robust estimations is to employ a combination of the nonsmooth quantile regression loss and nuclear-norm regularizer. Nevertheless, directly applying existing techniques may result in slow convergence rates due to the <i>doubly</i> nonsmooth objective. To expedite the computation process, a decentralized surrogate matrix quantile regression method is proposed in this article. The proposed algorithm has a simple implementation and can provably converge at a linear rate. Additionally, we provide a statistical guarantee that our estimate can achieve an almost optimal convergence rate, regardless of the number of nodes. Numerical simulations confirm the efficacy of our approach.