Risk bounds for quantile trend filtering

• 成果类型: Article
• 署名作者: Padilla, Oscar Hernan Madrid; Chatterjee, Sabyasachi
• 署名单位: University of California System; University of California Los Angeles; University of Illinois System; University of Illinois Urbana-Champaign
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asab045
• 发表日期: 2022
• 页码: 751768
• 关键词: fused lasso; regression; splines; asymptotics; adaptation; algorithm

摘要

We study quantile trend filtering, a recently proposed method for nonparametric quantile regression, with the goal of generalizing existing risk bounds for the usual trend-filtering estimators that perform mean regression. We study both the penalized and the constrained versions, of order r >= 1, of univariate quantile trend filtering. Our results show that both the constrained and the penalized versions of order r >= 1 attain the minimax rate up to logarithmic factors, when the (r - 1)th discrete derivative of the true vector of quantiles belongs to the class of bounded-variation signals. Moreover, we 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 employ our proof techniques to obtain new results for multivariate quantile total-variation denoising and high-dimensional quantile linear regression.