{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,22]],"date-time":"2026-02-22T19:27:09Z","timestamp":1771788429781,"version":"3.50.1"},"reference-count":18,"publisher":"MIT Press","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Neural Computation"],"published-print":{"date-parts":[[2017,2]]},"abstract":"<jats:p>A cross-validation method based on [Formula: see text] replications of two-fold cross validation is called an [Formula: see text] cross validation. An [Formula: see text] cross validation is used in estimating the generalization error and comparing of algorithms\u2019 performance in machine learning. However, the variance of the estimator of the generalization error in [Formula: see text] cross validation is easily affected by random partitions. Poor data partitioning may cause a large fluctuation in the number of overlapping samples between any two training (test) sets in [Formula: see text] cross validation. This fluctuation results in a large variance in the [Formula: see text] cross-validated estimator. The influence of the random partitions on variance becomes serious as [Formula: see text] increases. Thus, in this study, the partitions with a restricted number of overlapping samples between any two training (test) sets are defined as a block-regularized partition set. The corresponding cross validation is called block-regularized [Formula: see text] cross validation ([Formula: see text] BCV). It can effectively reduce the influence of random partitions. We prove that the variance of the [Formula: see text] BCV estimator of the generalization error is smaller than the variance of [Formula: see text] cross-validated estimator and reaches the minimum in a special situation. An analytical expression of the variance can also be derived in this special situation. This conclusion is validated through simulation experiments. Furthermore, a practical construction method of [Formula: see text] BCV by a two-level orthogonal array is provided. Finally, a conservative estimator is proposed for the variance of estimator of the generalization error.<\/jats:p>","DOI":"10.1162\/neco_a_00923","type":"journal-article","created":{"date-parts":[[2016,12,28]],"date-time":"2016-12-28T22:26:01Z","timestamp":1482963961000},"page":"519-554","source":"Crossref","is-referenced-by-count":8,"title":["Block-Regularized<i>m<\/i>\u00d7<i>2<\/i>Cross-Validated Estimator of the Generalization Error"],"prefix":"10.1162","volume":"29","author":[{"given":"Ruibo","family":"Wang","sequence":"first","affiliation":[{"name":"School of Software, Shanxi University, Taiyuan 030006, P.R.C."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Software, Shanxi University, Taiyuan 030006, P.R.C."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jihong","family":"Li","sequence":"additional","affiliation":[{"name":"School of Software, Shanxi University, Taiyuan 030006, P.R.C."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xingli","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Software, Shanxi University, Taiyuan 030006, P.R.C."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jing","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Shanxi University, Taiyuan 030006, P.R.C."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"281","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1162\/089976699300016007"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1214\/09-SS054"},{"key":"B3","first-page":"1089","volume":"5","author":"Bengio Y.","year":"2004","journal-title":"Journal of Machine Learning Research"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1162\/089976698300017197"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9868.2011.01005.x"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9868.2008.00674.x"},{"key":"B7","author":"Friedman J.","year":"2001","journal-title":"The elements of statistical learning"},{"key":"B9","first-page":"1127","volume":"6","author":"Markatou M.","year":"2005","journal-title":"Journal of Machine Learning Research"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1976.10481534"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1023\/A:1024068626366"},{"issue":"6","key":"B12","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1111\/j.2517-6161.1996.tb02094.x","volume":"58","author":"Nason G.","year":"1996","journal-title":"Journal of the Royal Statistical Society, Series B"},{"key":"B13","first-page":"229","author":"Stani\u0161i\u0107 P.","year":"2012","journal-title":"Mediterranean Conference on Embedded Computing"},{"key":"B14","doi-asserted-by":"publisher","DOI":"10.1214\/08-AOAS224"},{"key":"B15","doi-asserted-by":"publisher","DOI":"10.1016\/j.patrec.2015.08.002"},{"key":"B16","doi-asserted-by":"publisher","DOI":"10.1162\/NECO_a_00532"},{"key":"B17","author":"Wu C. 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