Distributed Stochastic Smoothing ADMM for Penalized Quantile Regression

Quantile regression is well suited to heterogeneous and heavy-tailed data, but computation becomes challenging for large, distributed data sets because the check loss is nonsmooth. We propose a distributed stochastic smoothing alternating direction method of multipliers (DSS-ADMM) for horizontally partitioned penalized quantile regression. Each worker computes a mini-batch gradient of a Huber-smoothed check loss, and a coordinator performs a single proximal aggregation step for the regularizer. Raw observations remain local, worker updates run in parallel, and the method requires no matrix inversion. For proper, closed, and convex penalties, stacking the local coefficient vectors yields a standard two-block stochastic ADMM formulation. With fixed smoothing, we establish an expected $O(\log K/\sqrt K)$ joint objective-feasibility bound and an explicit $\eps/4$ approximation term for the original check-loss objective; when the smooth block is strongly convex, the bound improves to $O(\log K/K)$. We also characterize the scope of an extension to the minimax concave penalty and the smoothly clipped absolute deviation penalty. Reproducible simulations consider both homogeneous worker partitions, in which observations are independently and identically distributed across workers, and heterogeneous partitions, in which worker-specific covariate distributions differ. Sensitivity studies and analyses of the diabetes and Engel data illustrate the trade-offs among per-observation gradient evaluations, communication, consensus, sparsity, and prediction.