Distributed quantile regression over sensor networks via the primal–dual hybrid gradient algorithm

As one of the important statistical methods, quantile regression (QR) extends traditional regression analysis. In QR, various quantiles of the response variable are modeled as linear functions of the predictors, allowing for a more flexible analysis of how the predictors affect different parts of the response variable distribution. QR offers several advantages over standard linear regression due to its focus on estimating conditional quantiles rather than the conditional mean of the response variable. This paper investigates QR over sensor networks, where each node has access to a local dataset and collaboratively estimates a global QR model. QR solves a non-smooth optimization problem characterized by a piecewise linear loss function, commonly known as the check function. We reformulate this non-smooth optimization problem as the task of finding a saddle point of a convex–concave objective and develop a distributed primal–dual hybrid gradient (dPDHG) algorithm for this purpose. Theoretical analyses guarantee the convergence of the proposed algorithm under mild assumptions, while experimental results show that the dPDHG algorithm converges significantly faster than subgradient-based schemes.