Quantile regression is a powerful tool for learning the relationship between a response variable and a multivariate predictor while exploring heterogeneous effects. However, the non-smooth piecewise linear loss function introduces challenges to the computational aspect when the number of covariates is large. To address the aforementioned challenge, we propose a convolution-type smoothing approach that turns the non-differentiable quantile piecewise linear loss function into a twice- differentiable, globally convex, and locally strongly convex surrogate, which admits a fast and scalable gradient-based algorithm to perform optimization. In the low-dimensional setting, we establish nonasymptotic error bounds for the resulting smoothed estimator. In the high-dimensional setting, we propose the concave regularized smoothed quantile regression estimator, which we solve using a multi-stage convex relaxation algorithm. Theoretically, we characterize both the algorithmic error due to non-convexity and statistical error for the resulting estimator simultaneously. We show that running the multi-stage algorithm for a few iterations will yield an estimator that achieves the oracle property. Our results suggest that the smoothing approach leads to a significant computational gain without a loss in statistical accuracy.
Kean Ming Tan is currently an assistant professor at the Department of Statistics at University of Michigan. He obtained his PhD from the Department of Biostatistics at the University of Washington. His primary research interest is in developing statistical methodology for analyzing complex large-scale data.