{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T13:10:14Z","timestamp":1772284214785,"version":"3.50.1"},"reference-count":38,"publisher":"MIT Press","issue":"10","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Neural Computation"],"published-print":{"date-parts":[[2014,10]]},"abstract":"<jats:p>Regularization is a well-recognized powerful strategy to improve the performance of a learning machine and l<jats:sup>q<\/jats:sup>regularization schemes with [Formula: see text] are central in use. It is known that different q leads to different properties of the deduced estimators, say, l<jats:sup>2<\/jats:sup>regularization leads to a smooth estimator, while l<jats:sup>1<\/jats:sup>regularization leads to a sparse estimator. Then how the generalization capability of l<jats:sup>q<\/jats:sup>regularization learning varies with q is worthy of investigation. In this letter, we study this problem in the framework of statistical learning theory. Our main results show that implementing l<jats:sup>q<\/jats:sup>coefficient regularization schemes in the sample-dependent hypothesis space associated with a gaussian kernel can attain the same almost optimal learning rates for all [Formula: see text]. That is, the upper and lower bounds of learning rates for l<jats:sup>q<\/jats:sup>regularization learning are asymptotically identical for all [Formula: see text]. Our finding tentatively reveals that in some modeling contexts, the choice of q might not have a strong impact on the generalization capability. From this perspective, q can be arbitrarily specified, or specified merely by other nongeneralization criteria like smoothness, computational complexity or sparsity.<\/jats:p>","DOI":"10.1162\/neco_a_00641","type":"journal-article","created":{"date-parts":[[2014,7,24]],"date-time":"2014-07-24T15:06:10Z","timestamp":1406214370000},"page":"2350-2378","source":"Crossref","is-referenced-by-count":13,"title":["Learning Rates of<i>l<sup>q<\/sup><\/i>Coefficient Regularization Learning with Gaussian Kernel"],"prefix":"10.1162","volume":"26","author":[{"given":"Shaobo","family":"Lin","sequence":"first","affiliation":[{"name":"Institute for Information and System Sciences, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, P.R.C."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jinshan","family":"Zeng","sequence":"additional","affiliation":[{"name":"Institute for Information and System Sciences, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, P.R.C., and Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing, 100038, P.R.C."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Fang","sequence":"additional","affiliation":[{"name":"Institute for Information and System Sciences, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, P.R.C."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zongben","family":"Xu","sequence":"additional","affiliation":[{"name":"Institute for Information and System Sciences, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, P.R.C."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"281","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1214\/009053607000000631"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1214\/009053607000000839"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-006-0196-8"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-01-00923-5"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511618796"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20042"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20303"},{"key":"B8","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02888-9"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-004-0158-6"},{"key":"B10","first-page":"1539","volume-title":"Advances in neural information processing systems 24","author":"Eberts M.","year":"2011"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2011.05.034"},{"key":"B12","doi-asserted-by":"publisher","DOI":"10.1007\/b97848"},{"key":"B13","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-21606-5"},{"key":"B14","doi-asserted-by":"publisher","DOI":"10.1142\/S0219530511001923"},{"key":"B16","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2013.2265397"},{"key":"B18","doi-asserted-by":"publisher","DOI":"10.1214\/09-AOS728"},{"key":"B19","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-009-9080-0"},{"key":"B20","doi-asserted-by":"crossref","DOI":"10.7551\/mitpress\/4175.001.0001","volume-title":"Learning with kernel: Support vector machine, regularization, optimization, and beyond","author":"Sch\u00f6lkopf B.","year":"2001"},{"key":"B21","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2011.01.001"},{"key":"B22","doi-asserted-by":"publisher","DOI":"10.1162\/NECO_a_00178"},{"key":"B23","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2012.03.009"},{"key":"B24","volume-title":"Support vector machines","author":"Steinwart I.","year":"2008"},{"key":"B25","volume-title":"Proceedings of the 22nd Conference on Learning Theory","author":"Steinwart I.","year":"2009"},{"key":"B26","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2006.881713"},{"key":"B27","doi-asserted-by":"publisher","DOI":"10.1214\/009053606000001226"},{"key":"B28","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2010.04.001"},{"key":"B29","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"Tibshirani R.","year":"1995","journal-title":"J. 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