Byzantine-robust distributed sparse learning for M-estimation

In a distributed computing environment, there is usually a small fraction of machines that are corrupted and send arbitrary erroneous information to the master machine. This phenomenon is modeled as a Byzantine failure. Byzantine-robust distributed learning has recently become an important topic in machine learning research. In this paper, we develop a Byzantine-resilient method for the distributed sparse M-estimation problem. When the loss function is non-smooth, it is computationally costly to solve the penalized non-smooth optimization problem in a direct manner. To alleviate the computational burden, we construct a pseudo-response variable and transform the original problem into an $$\ell _1$$ -penalized least-squares problem, which is much more computationally feasible. Based on this idea, we develop a communication-efficient distributed algorithm. Theoretically, we show that the proposed estimator obtains a fast convergence rate with only a constant number of iterations. Furthermore, we establish a support recovery result, which, to the best of our knowledge, is the first such result in the literature of Byzantine-robust distributed learning. We demonstrate the effectiveness of our approach in simulation.