{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:08:11Z","timestamp":1760242091182,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2018,12,14]],"date-time":"2018-12-14T00:00:00Z","timestamp":1544745600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11671307,11571078"],"award-info":[{"award-number":["11671307,11571078"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Hubei Province in China","award":["2017CFB523"],"award-info":[{"award-number":["2017CFB523"]}]},{"name":"the Special Fund for Basic Scientific Research of Central Colleges, South-Central University for Nationalities","award":["CZY18033"],"award-info":[{"award-number":["CZY18033"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The minimum error entropy principle (MEE) is an alternative of the classical least squares for its robustness to non-Gaussian noise. This paper studies the gradient descent algorithm for MEE with a semi-supervised approach and distributed method, and shows that using the additional information of unlabeled data can enhance the learning ability of the distributed MEE algorithm. Our result proves that the mean squared error of the distributed gradient descent MEE algorithm can be minimax optimal for regression if the number of local machines increases polynomially as the total datasize.<\/jats:p>","DOI":"10.3390\/e20120968","type":"journal-article","created":{"date-parts":[[2018,12,14]],"date-time":"2018-12-14T04:44:42Z","timestamp":1544762682000},"page":"968","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Semi-Supervised Minimum Error Entropy Principle with Distributed Method"],"prefix":"10.3390","volume":"20","author":[{"given":"Baobin","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ting","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,12,14]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Principe, J.C. (2010). Renyi\u2019s entropy and Kernel perspectives. Information Theoretic Learning, Springer.","key":"ref_1","DOI":"10.1007\/978-1-4419-1570-2"},{"unstructured":"Erdogmus, D., and Principe, J.C. (2000). Comparison of entropy and mean square error criteria in adaptive system training using higher order statistics. Proceedings of the International Conference on ICA and Signal Separation, Springer.","key":"ref_2"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/S0925-2312(02)00526-X","article-title":"Blind source separation using Renyi\u2019s \u03b1-marginal entropies","volume":"49","author":"Erdogmus","year":"2002","journal-title":"Neurocomputing"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1966","DOI":"10.1109\/TSP.2003.812843","article-title":"Convergence properties and data efficiency of the minimum error entropy criterion in adaline training","volume":"51","author":"Erdogmus","year":"2003","journal-title":"IEEE Trans. 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