{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T12:42:56Z","timestamp":1769863376974,"version":"3.49.0"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2024,9,26]],"date-time":"2024-09-26T00:00:00Z","timestamp":1727308800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,9,26]],"date-time":"2024-09-26T00:00:00Z","timestamp":1727308800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2024,12]]},"DOI":"10.1007\/s11222-024-10506-5","type":"journal-article","created":{"date-parts":[[2024,9,26]],"date-time":"2024-09-26T15:42:32Z","timestamp":1727365352000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Support vector machine in big data: smoothing strategy and adaptive distributed inference"],"prefix":"10.1007","volume":"34","author":[{"given":"Kangning","family":"Wang","sequence":"first","affiliation":[]},{"given":"Jin","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Xiaofei","family":"Sun","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,9,26]]},"reference":[{"key":"10506_CR1","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1214\/009053607000000839","volume":"36","author":"G Blanchard","year":"2008","unstructured":"Blanchard, G., Bousquet, O., Massart, P.: Statistical performance of support vector machines. Ann. Stat. 36, 489\u2013531 (2008)","journal-title":"Ann. Stat."},{"key":"10506_CR2","doi-asserted-by":"crossref","first-page":"1352","DOI":"10.1214\/17-AOS1587","volume":"46","author":"H Battey","year":"2018","unstructured":"Battey, H., Fan, J., Liu, H., Lu, J., Zhu, Z.: Distributed testing and estimation under sparse high dimensional models. Ann. Stat. 46, 1352\u20131382 (2018)","journal-title":"Ann. Stat."},{"key":"10506_CR3","first-page":"1369","volume":"9","author":"K Chang","year":"2008","unstructured":"Chang, K., Hsieh, C., Lin, C.: Coordinate descent method for large scale $$l_{2}$$-loss linear support vector machines. J. Mach. Learn. Res. 9, 1369\u20131398 (2008)","journal-title":"J. Mach. Learn. Res."},{"key":"10506_CR4","volume":"144","author":"L Chen","year":"2020","unstructured":"Chen, L., Zhou, Y.: Quantile regression in big data: a divide and conquer based strategy. Comput. Stat. Data Anal. 144, 106892 (2020)","journal-title":"Comput. Stat. Data Anal."},{"key":"10506_CR5","first-page":"3244","volume":"47","author":"X Chen","year":"2019","unstructured":"Chen, X., Liu, W., Zhang, Y.: Quantile regression under memory constraint. Ann. Stat. 47, 3244\u20133273 (2019)","journal-title":"Ann. Stat."},{"key":"10506_CR6","first-page":"1655","volume":"24","author":"X Chen","year":"2014","unstructured":"Chen, X., Xie, M.: A split-and-conquer approach for analysis of extraordinarily large data. Stat. Sin. 24, 1655\u20131684 (2014)","journal-title":"Stat. Sin."},{"key":"10506_CR7","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1137\/S1052623495280615","volume":"7","author":"B Chen","year":"1997","unstructured":"Chen, B., Harker, P.: Smooth approximations to nonlinear complementarity problems. SIAM J. Optimiz. 7, 403\u2013420 (1997)","journal-title":"SIAM J. Optimiz."},{"key":"10506_CR8","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/BF01592244","volume":"71","author":"C Chen","year":"1995","unstructured":"Chen, C., Mangasarian, O.: Smoothing methods for convex inequalities and linear complementarity problems. Math. Program. 71, 51\u201369 (1995)","journal-title":"Math. Program."},{"key":"10506_CR9","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1137\/S0363012997315907","volume":"37","author":"X Chen","year":"1999","unstructured":"Chen, X., Ye, Y.: On homotopy-smoothing methods for variational inequalities. SIAM J. Control. Optim. 37, 589\u2013616 (1999)","journal-title":"SIAM J. Control. Optim."},{"key":"10506_CR10","first-page":"273","volume":"20","author":"C Cortes","year":"1995","unstructured":"Cortes, C., Vapnik, V.: Support-vector networks. Mach. Learn. 20, 273\u2013297 (1995)","journal-title":"Mach. Learn."},{"key":"10506_CR11","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/j.neucom.2019.10.118","volume":"408","author":"J Cervantes","year":"2020","unstructured":"Cervantes, J., Garcia-Lamont, F., Rodriguez-Mazahua, L., Lopez, A.: A comprehensive survey on support vector machine classification: applications, challenges and trends. Neurocomputing 408, 189\u2013215 (2020)","journal-title":"Neurocomputing"},{"key":"10506_CR12","unstructured":"Fan, J., Guo, Y., Wang, K.: Communication-efficient accurate statistical estimation, (2019). arXiv: 1906.04870"},{"key":"10506_CR13","unstructured":"Fan, J., Wang, D., Wang, K., Zhu, Z.: Distributed estimation of principal eigenspaces, (2017). arXiv: 1702.06488"},{"key":"10506_CR14","first-page":"1871","volume":"9","author":"R Fan","year":"2008","unstructured":"Fan, R., Chang, K., Hsieh, C., Wang, X., Lin, C.: LIBLINEAR: a library for large linear classification. J. Mach. Learn. Res. 9, 1871\u20131874 (2008)","journal-title":"J. Mach. Learn. Res."},{"key":"10506_CR15","unstructured":"Gopal, S., Yang, Y.: Distributed training of large-scale logistic models. In: International Conference on Machine Learning, 289\u2013297 (2013)"},{"key":"10506_CR16","doi-asserted-by":"crossref","first-page":"1327","DOI":"10.2307\/2999619","volume":"66","author":"J Horowitz","year":"1998","unstructured":"Horowitz, J.: Bootstrap methods for median regression models. Econometrica 66, 1327\u20131351 (1998)","journal-title":"Econometrica"},{"key":"10506_CR17","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/s10107-019-01369-0","volume":"174","author":"C Huang","year":"2019","unstructured":"Huang, C., Huo, X.: A distributed one-step estimator. Math. Program. 174, 41\u201376 (2019)","journal-title":"Math. Program."},{"key":"10506_CR18","doi-asserted-by":"crossref","first-page":"668","DOI":"10.1080\/01621459.2018.1429274","volume":"14","author":"MI Jordan","year":"2019","unstructured":"Jordan, M.I., Lee, J.D., Yang, Y.: Communication-efficient distributed statistical inference. J. Am. Stat. Assoc. 14, 668\u2013681 (2019)","journal-title":"J. Am. Stat. Assoc."},{"key":"10506_CR19","first-page":"1343","volume":"9","author":"J Koo","year":"2008","unstructured":"Koo, J., Lee, Y., Kim, Y., Park, C.: A bahadur representation of the linear support vector machine. J. Mach. Learn. Res. 9, 1343\u20131368 (2008)","journal-title":"J. Mach. Learn. Res."},{"key":"10506_CR20","doi-asserted-by":"crossref","first-page":"33","DOI":"10.2307\/1913643","volume":"46","author":"R Koenker","year":"1978","unstructured":"Koenker, R., Bassett, G.: Regression quantiles. Econometrica 46, 33\u201350 (1978)","journal-title":"Econometrica"},{"key":"10506_CR21","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511754098","volume-title":"Quantile regression","author":"R Koenker","year":"2005","unstructured":"Koenker, R.: Quantile regression. Cambridge University Press, Cambridge (2005)"},{"key":"10506_CR22","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1023\/A:1011215321374","volume":"20","author":"Y Lee","year":"2001","unstructured":"Lee, Y., Mangasarian, O.: SSVM: a smooth support vector machine for classification. Comput. Optim. Appl. 20, 5\u201322 (2001)","journal-title":"Comput. Optim. Appl."},{"key":"10506_CR23","first-page":"929","volume":"25","author":"O Lepski","year":"1997","unstructured":"Lepski, O., Mammen, E., Spokoiny, V.: Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors. Ann. Stat. 25, 929\u2013947 (1997)","journal-title":"Ann. Stat."},{"key":"10506_CR24","first-page":"6691","volume":"18","author":"H Lian","year":"2017","unstructured":"Lian, H., Fan, Z.: Divide-and-conquer for debiased l1-norm support vector machine in ultra-high dimensions. J. Mach. Learn. Res. 18, 6691\u20136716 (2017)","journal-title":"J. Mach. Learn. Res."},{"key":"10506_CR25","doi-asserted-by":"crossref","first-page":"6380","DOI":"10.1016\/j.csda.2007.02.006","volume":"51","author":"Y Liu","year":"2007","unstructured":"Liu, Y., Zhang, H., Park, C., Ahn, J.: Support vector machines with adaptive $$l_{q}$$ penalty. Comput. Stat. Data Anal. 51, 6380\u20136394 (2007)","journal-title":"Comput. Stat. Data Anal."},{"key":"10506_CR26","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1111\/rssb.12352","volume":"82","author":"L Luo","year":"2020","unstructured":"Luo, L., Song, P.: Renewable estimation and incremental inference in generalized linear models with streaming data sets. J. Roy. Stat. Soc. B 82, 69\u201397 (2020)","journal-title":"J. Roy. Stat. Soc. B"},{"key":"10506_CR27","volume-title":"Problem complexity and method efficiency in optimization","author":"A Nemirovski","year":"1983","unstructured":"Nemirovski, A., Yudin, D.: Problem complexity and method efficiency in optimization. Wiley, New York (1983)"},{"key":"10506_CR28","first-page":"1","volume":"17","author":"B Peng","year":"2016","unstructured":"Peng, B., Wang, L., Wu, Y.: An error bound for $$l_{1}$$-norm support vector machine coefficients in ultra-high dimension. J. Mach. Learn. Res. 17, 1\u201326 (2016)","journal-title":"J. Mach. Learn. Res."},{"key":"10506_CR29","doi-asserted-by":"crossref","first-page":"1691","DOI":"10.1080\/07350015.2021.1961789","volume":"40","author":"R Pan","year":"2022","unstructured":"Pan, R., Ren, T., Guo, B., Li, F., Li, G., Wang, H.: A note on distributed quantile regression by pilot sampling and one-step updating. J. Bus. Econ. Stat. 40, 1691\u20131700 (2022)","journal-title":"J. Bus. Econ. Stat."},{"key":"10506_CR30","doi-asserted-by":"crossref","first-page":"2257","DOI":"10.1016\/j.jspi.2012.03.002","volume":"142","author":"C Park","year":"2012","unstructured":"Park, C., Kim, K., Myung, R., Koo, J.: Oracle properties of scad-penalized support vector machine. J. Stat. Plan. Inference 142, 2257\u20132270 (2012)","journal-title":"J. Stat. Plan. Inference"},{"key":"10506_CR31","first-page":"575","volume":"35","author":"J Scovel","year":"2007","unstructured":"Scovel, J., Steinwart, I.: Fast rates for support vector machines using gaussian kernels. Ann. Stat. 35, 575\u2013607 (2007)","journal-title":"Ann. Stat."},{"key":"10506_CR32","first-page":"567","volume":"14","author":"S Shalev-Shwartz","year":"2013","unstructured":"Shalev-Shwartz, S., Zhang, T.: Stochastic dual coordinate ascent methods for regularized loss. J. Mach. Learn. Res. 14, 567\u2013599 (2013)","journal-title":"J. Mach. Learn. Res."},{"key":"10506_CR33","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s10107-014-0839-0","volume":"155","author":"S Shalev-Shwartz","year":"2016","unstructured":"Shalev-Shwartz, S., Zhang, T.: Accelerated proximal stochastic dual coordinate ascent for regularized loss minimization. Math. Program. 155, 105\u2013145 (2016)","journal-title":"Math. Program."},{"key":"10506_CR34","first-page":"1000","volume":"32","author":"O Shamir","year":"2014","unstructured":"Shamir, O., Srebro, N., Zhang, T.: Communication-efficient distributed optimization using an approximate newton-type method. Int Conf. Mach. Learn. 32, 1000\u20131008 (2014)","journal-title":"Int Conf. Mach. Learn."},{"key":"10506_CR35","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1109\/TIT.2004.839514","volume":"51","author":"I Steinwart","year":"2005","unstructured":"Steinwart, I.: Consistency of support vector machines and other regularized kernel machines. IEEE Trans. Inf. Theory 51, 128\u2013142 (2005)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"10506_CR36","doi-asserted-by":"crossref","unstructured":"Sun, G., Wang, X., Yan, Y., Zhang, R.: Statistical inference and distributed implementation for linear multi-category SVM. Stat 12, e611 (2023)","DOI":"10.1002\/sta4.611"},{"key":"10506_CR37","volume-title":"The nature of statistical learning theory","author":"V Vapnik","year":"1996","unstructured":"Vapnik, V.: The nature of statistical learning theory. Springer, New York (1996)"},{"key":"10506_CR38","first-page":"1","volume":"20","author":"X Wang","year":"2019","unstructured":"Wang, X., Yang, Z., Chen, X., Liu, W.: Distributed inference for linear support vector machine. J. Mach. Learn. Res. 20, 1\u201341 (2019)","journal-title":"J. Mach. Learn. Res."},{"key":"10506_CR39","doi-asserted-by":"crossref","DOI":"10.1016\/j.csda.2021.107265","volume":"162","author":"F Wang","year":"2021","unstructured":"Wang, F., Zhu, Y., Huang, D., Qi, H., Wang, H.: Distributed one-step upgraded estimation for non-uniformly and non-randomly distributed data. Comput. Stat. Data Anal. 162, 107265 (2021)","journal-title":"Comput. Stat. Data Anal."},{"key":"10506_CR40","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s11222-023-10247-x","volume":"33","author":"K Wang","year":"2023","unstructured":"Wang, K., Li, S.: Distributed statistical optimization for non-randomly stored big data with application to penalized learning. Stat. Comput. 33, 73 (2023)","journal-title":"Stat. Comput."},{"key":"10506_CR41","volume":"235","author":"K Wang","year":"2022","unstructured":"Wang, K., Wang, H., Li, S.: Renewable quantile regression for streaming datasets. Knowl.-Based Syst. 235, 107675 (2022)","journal-title":"Knowl.-Based Syst."},{"key":"10506_CR42","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.neucom.2019.11.010","volume":"387","author":"G Wang","year":"2020","unstructured":"Wang, G., Zhang, G., Choi, K., Lam, K., Lu, J.: Output based transfer learning with least squares support vector machine and its application in bladder cancer prognosis. Neurocomputing 387, 279\u2013292 (2020)","journal-title":"Neurocomputing"},{"key":"10506_CR43","volume":"155","author":"K Wang","year":"2024","unstructured":"Wang, K., Yang, J., Polat, K., Alhudhaif, A., Sun, X.: Convolution smoothing and non-convex regularization for support vector machine in high dimensions. Appl. Soft Comput. 155, 111433 (2024)","journal-title":"Appl. Soft Comput."},{"key":"10506_CR44","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1214\/aos\/1079120130","volume":"32","author":"T Zhang","year":"2004","unstructured":"Zhang, T.: Statistical behavior and consistency of classification methods based on convex risk minimization. Ann. Stat. 32, 56\u201384 (2004)","journal-title":"Ann. Stat."},{"key":"10506_CR45","first-page":"3299","volume":"16","author":"Y Zhang","year":"2015","unstructured":"Zhang, Y., Duchi, J., Wainwright, M.: Divide and conquer kernel ridge regression: a distributed algorithm with minimax optimal rates. J. Mach. Learn. Res. 16, 3299\u20133340 (2015)","journal-title":"J. Mach. Learn. Res."},{"key":"10506_CR46","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1111\/rssb.12100","volume":"78","author":"X Zhang","year":"2016","unstructured":"Zhang, X., Wu, Y., Wang, L., Li, R.: Variable selection for support vector machine in moderately high dimensions. J. R. Stat. Soc. Ser. B 78, 53\u201376 (2016)","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"10506_CR47","unstructured":"Zhu, X., Li, F., Wang, H.: Least squares approximation for a distributed system, (2019). arXiv preprint arXiv: 1908.04904"},{"key":"10506_CR48","doi-asserted-by":"crossref","first-page":"1400","DOI":"10.1214\/15-AOS1410","volume":"44","author":"T Zhao","year":"2016","unstructured":"Zhao, T., Cheng, G., Liu, H.: A partially linear framework for massive heterogeneous data. Ann. Stat. 44, 1400\u20131437 (2016)","journal-title":"Ann. Stat."}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-024-10506-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-024-10506-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-024-10506-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T12:11:32Z","timestamp":1732536692000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-024-10506-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,26]]},"references-count":48,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["10506"],"URL":"https:\/\/doi.org\/10.1007\/s11222-024-10506-5","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,26]]},"assertion":[{"value":"3 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 September 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 September 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"188"}}