Distributed quantile regression via convolution smoothing and optimal subsampling

With modern data increasingly collected and stored across computing nodes by design, such as by time, geography, or client, distributed quantile regression must confront non-randomly stored data, non-smooth objectives, and communication bottlenecks all at once. We propose two communication-efficient distributed quantile regression (QR) estimators that address these challenges jointly. First, we implement convolution smoothing to render the check loss differentiable and construct a Poisson subsampling-based estimator within the smoothed QR framework. Theoretically, we establish its convergence rate and asymptotic normality and derive L-optimal sampling probabilities by minimizing the trace of the asymptotic covariance, ensuring statistical efficiency. Building on this foundation, we introduce a Distributed Smoothed Quantile Regression estimator with Poisson subsampling (DSQR-P), in which each worker transmits only a small Poisson subsample and the associated gradient information to the master node, achieving substantial communication reduction and computational scalability while preserving near full-sample accuracy. For high-dimensional data, we further develop a regularized distributed estimator (DSQRH-P) incorporating sparsity-inducing penalties. The resulting optimization problem is efficiently solved via the Local Adaptive Majorization-Minimization (LAMM) algorithm. We also establish its oracle property under standard sparsity conditions. Extensive simulation studies and two real data applications demonstrate that the proposed methods achieve excellent scalability, robustness, and statistical efficiency while substantially reducing computational and communication costs.