Smoothed Quantile Regression with Large-Scale Inference

TL;DR

This paper introduces 'conquer', a smoothed quantile regression method that enables scalable, accurate inference in high-dimensional, large-scale data settings, with theoretical guarantees and practical implementation.

Contribution

We develop a convolution-type smoothing approach for quantile regression that is computationally efficient and theoretically sound, suitable for large-scale, high-dimensional data.

Findings

Conquer achieves accurate approximation and inference for large-scale quantile regression.

The method's estimator is asymptotically normal under weaker conditions than traditional methods.

Numerical studies demonstrate the practical effectiveness of conquer in large datasets.

Abstract

Quantile regression is a powerful tool for learning the relationship between a response variable and a multivariate predictor while exploring heterogeneous effects. In this paper, we consider statistical inference for quantile regression with large-scale data in the "increasing dimension" regime. We provide a comprehensive and in-depth analysis of a convolution-type smoothing approach that achieves adequate approximation to computation and inference for quantile regression. This method, which we refer to as {\it{conquer}}, turns the non-differentiable quantile loss function into a twice-differentiable, convex and locally strongly convex surrogate, which admits a fast and scalable Barzilai-Borwein gradient-based algorithm to perform optimization, and multiplier bootstrap for statistical inference. Theoretically, we establish explicit non-asymptotic bounds on both estimation and Bahadur-Kiefer linearization errors, from which we show that the asymptotic normality of the conquer estimator holds under a weaker requirement on the number of the regressors than needed for conventional quantile regression. Moreover, we prove the validity of multiplier bootstrap confidence constructions. Our numerical studies confirm the conquer estimator as a practical and reliable approach to large-scale inference for quantile regression. Software implementing the methodology is available in the \texttt{R} package \texttt{conquer}.