Multiple-output composite quantile regression through an optimal transport lens

Xuzhi Yang, Tengyao Wang
Proceedings of Thirty Seventh Conference on Learning Theory, PMLR 247:5076-5122, 2024.

Abstract

Composite quantile regression has been used to obtain robust estimators of regression coefficients in linear models with good statistical efficiency. By revealing an intrinsic link between the composite quantile regression loss function and the Wasserstein distance from the residuals to the set of quantiles, we establish a generalization of the composite quantile regression to the multiple-output settings. Theoretical convergence rates of the proposed estimator are derived both under the setting where the additive error possesses only a finite $\ell$-th moment (for $\ell > 2$) and where it exhibits a sub-Weibull tail. In doing so, we develop novel techniques for analyzing the M-estimation problem that involves Wasserstein-distance in the loss. Numerical studies confirm the practical effectiveness of our proposed procedure.

Cite this Paper

BibTeX
@InProceedings{pmlr-v247-yang24a, title = {Multiple-output composite quantile regression through an optimal transport lens}, author = {Yang, Xuzhi and Wang, Tengyao}, booktitle = {Proceedings of Thirty Seventh Conference on Learning Theory}, pages = {5076--5122}, year = {2024}, editor = {Agrawal, Shipra and Roth, Aaron}, volume = {247}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--03 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v247/yang24a/yang24a.pdf}, url = {https://proceedings.mlr.press/v247/yang24a.html}, abstract = {Composite quantile regression has been used to obtain robust estimators of regression coefficients in linear models with good statistical efficiency. By revealing an intrinsic link between the composite quantile regression loss function and the Wasserstein distance from the residuals to the set of quantiles, we establish a generalization of the composite quantile regression to the multiple-output settings. Theoretical convergence rates of the proposed estimator are derived both under the setting where the additive error possesses only a finite $\ell$-th moment (for $\ell > 2$) and where it exhibits a sub-Weibull tail. In doing so, we develop novel techniques for analyzing the M-estimation problem that involves Wasserstein-distance in the loss. Numerical studies confirm the practical effectiveness of our proposed procedure.} }
Endnote
%0 Conference Paper %T Multiple-output composite quantile regression through an optimal transport lens %A Xuzhi Yang %A Tengyao Wang %B Proceedings of Thirty Seventh Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2024 %E Shipra Agrawal %E Aaron Roth %F pmlr-v247-yang24a %I PMLR %P 5076--5122 %U https://proceedings.mlr.press/v247/yang24a.html %V 247 %X Composite quantile regression has been used to obtain robust estimators of regression coefficients in linear models with good statistical efficiency. By revealing an intrinsic link between the composite quantile regression loss function and the Wasserstein distance from the residuals to the set of quantiles, we establish a generalization of the composite quantile regression to the multiple-output settings. Theoretical convergence rates of the proposed estimator are derived both under the setting where the additive error possesses only a finite $\ell$-th moment (for $\ell > 2$) and where it exhibits a sub-Weibull tail. In doing so, we develop novel techniques for analyzing the M-estimation problem that involves Wasserstein-distance in the loss. Numerical studies confirm the practical effectiveness of our proposed procedure.
APA
Yang, X. & Wang, T.. (2024). Multiple-output composite quantile regression through an optimal transport lens. Proceedings of Thirty Seventh Conference on Learning Theory, in Proceedings of Machine Learning Research 247:5076-5122 Available from https://proceedings.mlr.press/v247/yang24a.html.

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