Risk Bounds for Quantile Trend Filtering

TL;DR

This paper establishes risk bounds for univariate quantile trend filtering, demonstrating near minimax optimal rates under minimal assumptions, including heavy-tailed errors, and extends the techniques to multivariate and high-dimensional settings.

Contribution

It provides the first risk bounds for quantile trend filtering that hold under minimal error assumptions and extends the analysis to multivariate and high-dimensional quantile regression.

Findings

Both penalized and constrained quantile trend filtering attain near minimax rates.

Risk bounds hold under heavy-tailed errors without moment assumptions.

Techniques are applicable to multivariate and high-dimensional quantile regression.

Abstract

We study quantile trend filtering, a recently proposed method for nonparametric quantile regression with the goal of generalizing existing risk bounds known for the usual trend filtering estimators which perform mean regression. We study both the penalized and the constrained version (of order $r \geq 1$) of univariate quantile trend filtering. Our results show that both the constrained and the penalized version (of order $r \geq 1$) attain the minimax rate up to log factors, when the $(r-1)$th discrete derivative of the true vector of quantiles belongs to the class of bounded variation signals. Moreover we also show that if the true vector of quantiles is a discrete spline with a few polynomial pieces then both versions attain a near parametric rate of convergence. Corresponding results for the usual trend filtering estimators are known to hold only when the errors are sub-Gaussian. In contrast, our risk bounds are shown to hold under minimal assumptions on the error variables. In particular, no moment assumptions are needed and our results hold under heavy-tailed errors. Our proof techniques are general and thus can potentially be used to study other nonparametric quantile regression methods. To illustrate this generality we also employ our proof techniques to obtain new results for multivariate quantile total variation denoising and high dimensional quantile linear regression.