Distributed High-dimensional Regression Under a Quantile Loss Function
This addresses robust distributed estimation for high-dimensional data with heavy-tailed noise, representing an incremental improvement by adapting quantile loss to distributed settings.
The paper tackles high-dimensional linear regression with heavy-tailed noise by using quantile regression loss, achieving a near-oracle convergence rate and support recovery guarantees without restrictions on machine count.
This paper studies distributed estimation and support recovery for high-dimensional linear regression model with heavy-tailed noise. To deal with heavy-tailed noise whose variance can be infinite, we adopt the quantile regression loss function instead of the commonly used squared loss. However, the non-smooth quantile loss poses new challenges to high-dimensional distributed estimation in both computation and theoretical development. To address the challenge, we transform the response variable and establish a new connection between quantile regression and ordinary linear regression. Then, we provide a distributed estimator that is both computationally and communicationally efficient, where only the gradient information is communicated at each iteration. Theoretically, we show that, after a constant number of iterations, the proposed estimator achieves a near-oracle convergence rate without any restriction on the number of machines. Moreover, we establish the theoretical guarantee for the support recovery. The simulation analysis is provided to demonstrate the effectiveness of our method.