Private Conditional Quantile Functions through Shared Location Learning

Conditional quantile functions support prediction at uncertainty levels chosen after training. A direct approach under differential privacy fits separate models at several levels. Training these models on the same records divides a fixed privacy budget among them, increasing the noise required for each fit. In this study, we propose DP-ShareQ, which combines a shared input-dependent location with a learned shift for each quantile level. We develop two learners for this representation. Joint learns the location and these offsets together, using quantile ordering to calibrate gradient noise. CDF first learns a private location predictor, then privately estimates its residual distribution and uses its quantiles as offsets. For both learners, we derive prediction bounds that quantify the approximation cost of shared quantile spacing and the error of private learning. Across six datasets with public linear and additive features, Joint and CDF reduce average normalized integrated pinball loss relative to output-perturbed separate fits by 25.9–30.3% and 31.0–31.5%, respectively, across privacy budgets \(\varepsilon\in{1,2,4}\).