Finite-Sample Distribution Theory and Efficient Large-Scale Inference for Online Quantile Regression

This paper studies online quantile regression for large-scale and streaming data using Stochastic SubGradient Descent (SSGD) with constant learning rates. Classical offline inference for quantile regression is computationally and memory intensive. Existing works of online inference for quantile regression provide only asymptotic guarantees and typically require sub-exponential tail conditions for distribution theory. To bridge these gaps, we introduce new techniques to prove a quenched central limit theorem (CLT) and finite-sample Gaussian approximation for SSGD under a finite-moment assumption. We further show that Ruppert-Polyak averaging with a constant learning rate has a non-vanishing bias and fails to satisfy CLT centering at the population target. Hence we propose suffix averaging to address this issue and establish its finite-sample Gaussian approximation. Based on these results, we provide an efficient online inference method for quantile regression that avoids covariance estimation. Numerical experiments show that our method achieves desirable empirical coverage rates and competitive performance compared to other inference methods. We also apply our approach to U.S. wage data to demonstrate its practical effectiveness.