# Cross Validation for Penalized Quantile Regression with a Case-Weight Adjusted Solution Path

## TL;DR

This paper introduces an efficient algorithm for exact leave-one-out cross validation in penalized quantile regression, overcoming limitations of approximation methods especially for extreme quantiles.

## Contribution

It develops a case-weight adjusted solution path algorithm using homotopy techniques to compute leave-one-out estimators efficiently for penalized quantile regression.

## Key findings

- Exact LOOCV scores can be computed efficiently from the full-data solution.
- Case influence varies with quantiles, data size, and penalty parameters.
- The method performs well in real-world applications.

## Abstract

Cross validation is widely used for selecting tuning parameters in regularization methods, but it is computationally intensive in general. To lessen its computational burden, approximation schemes such as generalized approximate cross validation (GACV) are often employed. However, such approximations may not work well when non-smooth loss functions are involved. As a case in point, approximate cross validation schemes for penalized quantile regression do not work well for extreme quantiles. In this paper, we propose a new algorithm to compute the leave-one-out cross validation scores exactly for quantile regression with ridge penalty through a case-weight adjusted solution path. Resorting to the homotopy technique in optimization, we introduce a case weight for each individual data point as a continuous embedding parameter and decrease the weight gradually from one to zero to link the estimators based on the full data and those with a case deleted. This allows us to design a solution path algorithm to compute all leave-one-out estimators very efficiently from the full-data solution. We show that the case-weight adjusted solution path is piecewise linear in the weight parameter, and using the solution path, we examine case influences comprehensively and observe that different modes of case influences emerge, depending on the specified quantiles, data dimensions and penalty parameter. We further illustrate the utility of the proposed algorithm in real-world applications.