Differentially Private Quantile Regression

Quantile regression (QR) is a powerful and robust statistical modeling method broadly used in many fields such as economics, ecology, and healthcare. However, it has not been well-explored in differential privacy (DP) since its loss function lacks strong convexity and twice differentiability, often required by many DP mechanisms. We implement the smoothed QR loss via convolution within the K-Norm Gradient mechanism (KNG) and prove the resulting estimate converges to the non-private one asymptotically. Additionally, our work is the first to extensively investigate the empirical performance of DP smoothing QR under pure-, approximate- and concentrated-DP and four mechanisms, and cases commonly encountered in practice such as heavy-tailed and heteroscedastic data. We find that the Objective Perturbation Mechanism and KNG are the top performers across the simulated settings.