🤖 AI Summary
This work presents the first unbiased estimator and self-normalized inference procedure for online quantile regression under user-level ε-local differential privacy (LDP). To address the challenge that the server cannot directly access raw estimating equations, the authors propose a mechanism based on a finite-alphabet communication channel: each user uploads a single privatized contribution by combining support-aware randomized quantization with randomized response, and the server employs a public decoder to correct bias, reconstruct an unbiased gradient, and perform online inference via projected Polyak–Ruppert averaging. The method avoids Hessian computation and establishes asymptotic normality for pre-specified scalar contrasts. Theoretical analysis guarantees privacy, unbiasedness, consistency, and asymptotic normality. Empirical results demonstrate superior privacy–utility trade-offs compared to Laplace and high-dimensional exponential mechanisms, with performance approaching the non-private benchmark as the privacy budget increases.
📝 Abstract
We study estimation and inference for online quantile regression under a one-report user-level $\eps$-locally differentially private ($\eps$-LDP) protocol. The main difficulty is that the standard quantile-regression estimating-equation contribution couples covariates with a residual comparison, so a server that receives only privatized reports cannot form the usual online update. We address this by developing a finite-alphabet channel in which each user computes the contribution locally, applies support-aware stochastic quantization and randomized response to one selected-block category, and sends one report. A public decoder corrects the randomized-response distortion and reconstructs a server-side estimating-equation input with the correct conditional mean. These decoded inputs are then used in projected Polyak-Ruppert averaging. For fixed finite channel designs, we establish local privacy, decoder unbiasedness, consistency, asymptotic normality, and Hessian-free self-normalized inference for prespecified scalar contrasts. Simulations and a New York City taxi-trip illustration show that the private trajectory approaches the nonprivate online reference as the privacy budget grows and outperforms direct Laplace and face-exponential geometric releases in the reported regimes.