{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T20:44:43Z","timestamp":1784925883071,"version":"3.55.0"},"reference-count":46,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,25]],"date-time":"2025-05-25T00:00:00Z","timestamp":1748131200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12371476"],"award-info":[{"award-number":["12371476"]}]},{"name":"Key Program for Youth Innovation of the University of Science and Technology of China","award":["12371476"],"award-info":[{"award-number":["12371476"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper proposes a novel data-driven distributionally robust framework for conditional quantile prediction under the fixed design setting of the covariates, which we refer to as Sinkhorn distributionally robust conditional quantile prediction. We derive a convex programming dual reformulation of the proposed problem and further develop a conic optimization reformulation for the case with finite support. Our method\u2019s superior performance is demonstrated through several numerical experiments, highlighting its effectiveness in practical applications.<\/jats:p>","DOI":"10.3390\/e27060557","type":"journal-article","created":{"date-parts":[[2025,5,25]],"date-time":"2025-05-25T20:26:50Z","timestamp":1748204810000},"page":"557","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Sinkhorn Distributionally Robust Conditional Quantile Prediction with Fixed Design"],"prefix":"10.3390","volume":"27","author":[{"given":"Guohui","family":"Jiang","sequence":"first","affiliation":[{"name":"Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230052, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tiantian","family":"Mao","sequence":"additional","affiliation":[{"name":"Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230052, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"33","DOI":"10.2307\/1913643","article-title":"Regression Quantiles","volume":"46","author":"Koenker","year":"1978","journal-title":"Econometrica"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1111\/j.1467-6435.2008.00417.x","article-title":"The Impact of Trade Liberalisation on Economic Growth: Evidence from a Quantile Regression Analysis","volume":"61","author":"Foster","year":"2008","journal-title":"Kyklos"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1093\/pcmedi\/pbz007","article-title":"Quantile regression for survival data in modern cancer research: Expanding statistical tools for precision medicine","volume":"2","author":"Hong","year":"2019","journal-title":"Precis. Clin. Med."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1111\/j.1467-9876.2011.01025.x","article-title":"Spatiotemporal quantile regression for detecting distributional changes in environmental processes","volume":"61","author":"Reich","year":"2012","journal-title":"J. R. Stat. Soc. Ser. C Appl. Stat."},{"key":"ref_5","unstructured":"(2009, July 13). Basel Committee on Banking Supervision: Revisions to the Basel II Market Risk Framework. Available online: https:\/\/www.bis.org\/publ\/bcbs158.htm."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1287\/opre.2018.1757","article-title":"The big data newsvendor: Practical insights from machine learning","volume":"67","author":"Ban","year":"2019","journal-title":"Oper. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1080\/01621459.2018.1424632","article-title":"From fixed-x to random-x regression: Bias-variance decompositions, covariance penalties, and prediction error estimation","volume":"115","author":"Rosset","year":"2018","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1137\/S1052623499363220","article-title":"The sample average approximation method for stochastic discrete optimization","volume":"12","author":"Kleywegt","year":"2002","journal-title":"SIAM J. Optim."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1639","DOI":"10.1287\/mnsc.2020.3903","article-title":"Distributionally robust conditional quantile prediction with fixed-design","volume":"68","author":"Qi","year":"2022","journal-title":"Manag. Sci."},{"key":"ref_10","unstructured":"Arrow,  K.J., Karlin, S., and Scarf, H.E. (1958). Studies in the mathematical theory of inventory and production. A Min\u2013Max Solution of an Inventory Problem, Stanford University Press. Available online: https:\/\/www.rand.org\/pubs\/papers\/P910.html."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1287\/opre.1090.0741","article-title":"Distributionally robust optimization under moment uncertainty with application to data-driven problems","volume":"58","author":"Delage","year":"2010","journal-title":"Oper. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s10107-011-0494-7","article-title":"Distributionally robust joint chance constraints with second-order moment information","volume":"137","author":"Zymler","year":"2013","journal-title":"Math. Program."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1358","DOI":"10.1287\/opre.2014.1314","article-title":"Distributionally robust convex optimization","volume":"62","author":"Wiesemann","year":"2014","journal-title":"Oper. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1287\/moor.1040.0137","article-title":"A semidefinite programming approach to optimal-moment bounds for convex classes of distributions","volume":"30","author":"Popescu","year":"2005","journal-title":"Math. Oper. Res."},{"key":"ref_15","first-page":"271","article-title":"Generalized gauss inequalities via semidefinite programming","volume":"156","author":"Goulart","year":"2015","journal-title":"Math. Program."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1287\/mnsc.1080.0951","article-title":"Persistency problem and its applications in choice probleming","volume":"55","author":"Natarajan","year":"2009","journal-title":"Manag. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1287\/opre.1110.1011","article-title":"Price of correlations in stochastic optimization","volume":"60","author":"Agrawal","year":"2012","journal-title":"Oper. Res."},{"key":"ref_18","unstructured":"Hu, Z., and Hong, L.J. (2012, November 23). Kullback-Leibler Divergence Constrained Distributionally Robust Optimization. Available online: https:\/\/optimization-online.org\/?p=12225."},{"key":"ref_19","first-page":"341","article-title":"Robust solutions of optimization problems affected by uncertain probabilities","volume":"59","author":"Melenberg","year":"2013","journal-title":"Manag. Sci."},{"key":"ref_20","first-page":"1","article-title":"Data-driven stochastic programming using phi-divergences","volume":"11","author":"Bayraksan","year":"2015","journal-title":"INFORMS TutORials Oper. Res."},{"key":"ref_21","first-page":"130","article-title":"Wasserstein distributionally robust optimization: Theory and applications in machine learning","volume":"15","author":"Kuhn","year":"2019","journal-title":"INFORMS TutORials Oper. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1287\/moor.2018.0936","article-title":"Quantifying distributional problem risk via optimal transport","volume":"44","author":"Blanchet","year":"2019","journal-title":"Math. Oper. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1287\/moor.2022.1275","article-title":"Distributionally robust stochastic optimization with Wasserstein distance","volume":"48","author":"Gao","year":"2022","journal-title":"Math. Oper. Res."},{"key":"ref_24","unstructured":"Staib, M., and Jegelka, S. (2019, January 8\u201314). Distributionally robust optimization and generalization in kernel methods. Proceedings of the Advances in Neural Information Processing Systems, Vancouver, BC, Canada."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1214\/aoms\/1177703591","article-title":"A relationship between arbitrary positive matrices and doubly stochastic matrices","volume":"35","author":"Sinkhorn","year":"1964","journal-title":"Ann. Math. Stat."},{"key":"ref_26","unstructured":"Cuturi, M. (2013, January 5\u201310). Sinkhorn distances: Lightspeed computation of optimal transport. Proceedings of the 27th International Conference on Neural Information Processing Systems (NIPS\u201913), New York, NY, USA."},{"key":"ref_27","unstructured":"Wang, J., Gao, R., and Xie, Y. (2025, March 26). Sinkhorn Distributionally Robust Optimization. Available online: https:\/\/arxiv.org\/abs\/2109.11926."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1051\/cocv\/2023019","article-title":"Regularization for wasserstein distributionally robust optimization","volume":"29","author":"Azizian","year":"2023","journal-title":"ESAIM Control Optim. Calc. Var."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Courty, N., Flamary, R., and Tuia, D. (2014, January 15\u201319). Domain adaptation with regularized optimal transport. Proceedings of the Joint European Conference on Machine Learning and Knowledge Discovery in Databases, Nancy, France.","DOI":"10.1007\/978-3-662-44848-9_18"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1853","DOI":"10.1109\/TPAMI.2016.2615921","article-title":"Optimal transport for domain adaptation","volume":"39","author":"Courty","year":"2016","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_31","unstructured":"Luise, G., Rudi, A., Pontil, M., and Ciliberto, C. (2018, January 2\u20138). Differential properties of sinkhorn approximation for learning with wasserstein distance. Proceedings of the 32nd International Conference on Neural Information Processing Systems, Montr\u00e9al, QC, Canada."},{"key":"ref_32","unstructured":"Patrini, G., Van den Berg, R., Forre, P., Carioni, M., Bhargav, S., Welling, M., Genewein, T., and Nielsen, F. (2019, January 22\u201325). Sinkhorn Autoencoders. Proceedings of the 35th Uncertainty in Artificial Intelligence Conference, Tel Aviv, Israel."},{"key":"ref_33","unstructured":"Lin, T., Fan, C., Ho, N., Cuturi, M., and Jordan, M. (2020, January 6\u201312). Projection robust wasserstein distance and riemannian optimization. Proceedings of the 34th International Conference on Neural Information Processing Systems, Vancouver, BC, Canada."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Wang, J., Gao, R., and Xie, Y. (2021, January 12\u201320). Two-sample test using projected wasserstein distance. Proceedings of the 2021 IEEE International Symposium on Information Theory (ISIT), Melbourne, Australia.","DOI":"10.1109\/ISIT45174.2021.9518186"},{"key":"ref_35","unstructured":"Wang, J., Gao, R., and Xie, Y. (2022, January 28\u201330). Two-sample test with kernel projected wasserstein distance. Proceedings of the 25th International Conference on Artificial Intelligence and Statistics."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2103","DOI":"10.1137\/19M1284865","article-title":"Sample complexity of sample average approximation for conditional stochastic optimization","volume":"30","author":"Hu","year":"2023","journal-title":"SIAM J. Optim."},{"key":"ref_37","unstructured":"Cheng, B., and Xie, X. (2023). Distributionally Robust Conditional Quantile Prediction with Wasserstein Ball. JUSTC, accepted."},{"key":"ref_38","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, John Wiley & Sons. [2nd ed.]."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1111\/1467-9574.00056","article-title":"Conditioning as disintegration","volume":"51","author":"Joseph","year":"1997","journal-title":"Stat. Neerl."},{"key":"ref_40","unstructured":"Yang, Y., Zhou, Y., and Lu, Z. (2024, January 15). A Stochastic Algorithm for Sinkhorn Distance-Regularized Distributionally Robust Optimization. Proceedings of the OPT2024: 16th Annual Workshop on Optimization for Machine Learning, Vancouver, BC, Canada."},{"key":"ref_41","unstructured":"Grant, M., and Boyd, S. (2020, January 01). CVX:Matlab Software for Disciplined Convex Programming. Available online: https:\/\/cvxr.com\/cvx\/."},{"key":"ref_42","first-page":"1148","article-title":"Challenging the empirical mean and empirical variance: A deviation study","volume":"48","author":"Catoni","year":"2012","journal-title":"Ann. l\u2019IHP Probab. Stat."},{"key":"ref_43","first-page":"1","article-title":"Non-asymptotic guarantees for robust statistical learning under infinite variance assumption","volume":"24","author":"Xu","year":"2023","journal-title":"J. Mach. Learn. Res."},{"key":"ref_44","unstructured":"Xu, Y., Zhu, S., Yang, S., Zhang, C., Jin, R., and Yang, T. (, January 22\u201325). Learning with non-convex truncated losses by SGD. Proceedings of the Thirty-Fifth Conference on Uncertainty in Artificial Intelligence (UAI 2019), Tel Aviv, Israel."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1080\/10485252.2012.749257","article-title":"Fixed-design regression estimation based on real and artificial data","volume":"25","author":"Furer","year":"2013","journal-title":"J. Nonparametr. Stat."},{"key":"ref_46","unstructured":"Dua, D., and Graff, C. (2020, February 29). UCI Machine Learning Repository. Available online: https:\/\/archive.ics.uci.edu\/dataset\/560\/seoul+bike+sharing+demand."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/6\/557\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:40:14Z","timestamp":1760031614000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/6\/557"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,25]]},"references-count":46,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["e27060557"],"URL":"https:\/\/doi.org\/10.3390\/e27060557","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,25]]}}}