Smoothed Quantile Regression with Large-Scale Inference

Quantile regression is a powerful tool for learning the relationship between a scalar response and a multivariate predictor in the presence of heavier tails and/or data heterogeneity. In the present paper, we consider statistical inference for quantile regression with large-scale data in the increasing dimension regime. We provide a comprehensive study of a convolution-type smoothing approach to achieving an adequate approximation to computation and inference for quantile regression. The ensuing estimator, which we refer to as conquer, turns the non-differentiable quantile loss function into a twice-differentiable, globally convex, and locally strongly convex surrogate, which admits a fast and scalable Barzilai-Borwein gradient-based algorithm to perform optimization, and a Rademacher multiplier bootstrap method for statistical inference. In the theoretical investigations of the conquer estimator, we establish nonasymptotic error bounds on the Bahadur-Kiefer linearization, from which we show that the asymptotic normality of the smoothed quantile regression estimator holds under a weaker requirement on the dimension of the predictors than needed for the exact quantile regression estimator. Our numerical studies confirm the conquer estimator as a practical and reliable approach to large-scale inference for quantile regression.