Byzantine-robust and efficient distributed sparsity learning: a surrogate composite quantile regression approach

Distributed statistical learning has gained significant traction recently, mainly due to the availability of unprecedentedly massive datasets. The objective of distributed statistical learning is to learn models by effectively utilizing data scattered across various machines. However, its performance can be impeded by three significant challenges: arbitrary noises, high dimensionality, and machine failures—the latter being specifically referred to as Byzantine failure. To address the first two challenges, we propose leveraging the potential of composite quantile regression in conjunction with the \(\ell _1\) ${\ell }_1$ penalty. However, this combination introduces a doubly nonsmooth objective function, posing new challenges. In such scenarios, most existing Byzantine-robust methods exhibit slow sublinear convergence rates and fail to achieve near-optimal statistical convergence rates. To fill this gap, we introduce a novel smoothing procedure that effectively handles the nonsmooth aspects. This innovation allows us to develop a Byzantine-robust sparsity learning algorithm that converges provably to the near-optimal convergence rate linearly. Moreover, we establish support recovery guarantees for our proposed methods. We substantiate the effectiveness of our approaches through comprehensive empirical analyses.