Regularized Adaptive Huber Matrix Regression and Distributed Learning

Abstract

Matrix regression provides a powerful technique for analyzing matrixtype data, as exemplified by many contemporary applications.Despite the rapid advance, distributed learning for robust matrix regression to deal with heavytailed noises in the big data regime still remains untouched.In this paper, we first consider adaptive Huber matrix regression with a nuclear norm penalty, which enjoys insensitivity to heavy-tailed noises without losing the statistical accuracy.To further enhance the scalability in massive data applications, we employ the communication-efficient surrogate likelihood framework to develop distributed robust matrix regression, which can be efficiently implemented through the ADMM algorithms.Under only bounded (1 + δ)-th moment on the noise for some δ ∈ (0, 1], we provide upper bounds for the estimation error of the central estimator and the distributed estimator, and prove they can achieve the same rate as established with sub-Gaussian tails when only the second moment of noise exists.Numerical studies verify the advantage of the proposed method