Adaptive distributed smooth composite quantile regression estimation for large-scale data

Composite quantile regression (CQR) is a good statistical learning tool because of its estimation efficiency and robustness advantages, but the growing size of modern data is bringing challenges to it. First, the non-smoothness of CQR loss function poses high computation burden in large-scale problems. Second, although some distributed CQR algorithms have been proposed, they heavily rely on uniformity and randomness conditions, which are frequently violated in practice. To address these issues, this article first proposes a smooth CQR by constructing a smooth loss, which can converge to the original non-smooth loss uniformly. Then a distributed CQR is developed, in which the estimator can be calculated conveniently by minimizing a pilot sample-based distributed surrogate loss. In particular, it can be adaptive when the uniformity or randomness condition is violated. The established theoretical results and numerical experiments all confirm the proposed methods.