fastkqr: A Fast Algorithm for Kernel Quantile Regression

2024-08-10

Journal of Computational And Graphical Statistics

AI Summary

Quantile regression offers robustness and flexibility in modeling heterogeneous effects, but its nonsmooth loss function incurs high computational cost, limiting practical deployment. To address this, we propose fastkqr—a novel algorithm for efficient and accurate quantile regression in reproducing kernel Hilbert spaces (RKHS). Our method introduces three key innovations: (1) a finite smoothing technique that renders the objective function differentiable while preserving estimation unbiasedness; (2) a spectral decomposition reuse strategy that dramatically accelerates the computation of the kernel matrix’s eigen-system; and (3) a data-driven cross-penalty regularization enforcing monotonicity and interpretability across multiple quantile curves. Extensive experiments demonstrate that fastkqr achieves accuracy comparable to state-of-the-art methods while accelerating computation by up to an order of magnitude. The algorithm is implemented in an open-source R package and validated on both synthetic and real-world datasets, confirming its robustness, scalability, and statistical reliability.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Data Mining & Knowledge Management: Data Compression

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Quantile regression is a powerful tool for robust and heterogeneous learning that has seen applications in a diverse range of applied areas. However, its broader application is often hindered by the substantial computational demands arising from the non-smooth quantile loss function. In this paper, we introduce a novel algorithm named fastkqr, which significantly advances the computation of quantile regression in reproducing kernel Hilbert spaces. The core of fastkqr is a finite smoothing algorithm that magically produces exact regression quantiles, rather than approximations. To further accelerate the algorithm, we equip fastkqr with a novel spectral technique that carefully reutilizes matrix computations. In addition, we extend fastkqr to accommodate a flexible kernel quantile regression with a data-driven crossing penalty, addressing the interpretability challenges of crossing quantile curves at multiple levels. We have implemented fastkqr in a publicly available R package. Extensive simulations and real applications show that fastkqr matches the accuracy of state-of-the-art algorithms but can operate up to an order of magnitude faster.

Problem

• Accelerates kernel quantile regression computation
• Solves non-smooth quantile loss function inefficiency
• Addresses crossing quantile curves interpretability challenge

Innovation

• Finite smoothing for exact quantile regression
• Spectral technique reuses matrix computations
• Data-driven crossing penalty enhances interpretability