# Locally Private Online Quantile Regression: Estimation and Inference

arXiv: https://arxiv.org/abs/2607.05312 · HTML: https://mlsanity.com/paper/2607.05312/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 102 references that are in the map.
- First posted on arXiv 6 Jul 2026.

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## Abstract (arXiv)

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.