{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T03:34:45Z","timestamp":1769744085077,"version":"3.49.0"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2011,11,20]],"date-time":"2011-11-20T00:00:00Z","timestamp":1321747200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mach Learn"],"published-print":{"date-parts":[[2012,4]]},"DOI":"10.1007\/s10994-011-5269-0","type":"journal-article","created":{"date-parts":[[2011,11,19]],"date-time":"2011-11-19T15:55:12Z","timestamp":1321718112000},"page":"1-32","source":"Crossref","is-referenced-by-count":7,"title":["Online variance minimization"],"prefix":"10.1007","volume":"87","author":[{"given":"Manfred K.","family":"Warmuth","sequence":"first","affiliation":[]},{"given":"Dima","family":"Kuzmin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"5269_CR1","first-page":"437","volume-title":"Proceedings of the 21st annual conference on learning theory (COLT \u201908)","author":"J. Abernethy","year":"2008","unstructured":"Abernethy, J., Warmuth, M. K., & Yellin, J. (2008). When random play is optimal against an adversary. In Proceedings of the 21st annual conference on learning theory (COLT \u201908) (pp. 437\u2013445)."},{"key":"5269_CR2","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1145\/1143844.1143846","volume-title":"Proceedings of the 23rd international conference on machine learning (ICML\u00a0\u201906)","author":"A. Agarwal","year":"2006","unstructured":"Agarwal, A., Hazan, E., Kale, S., & Schapire, R.\u00a0E. (2006). Algorithms for portfolio management based on the Newton method. In Proceedings of the 23rd international conference on machine learning (ICML\u00a0\u201906) (pp. 9\u201316). New York: ACM. http:\/\/doi.acm.org\/10.1145\/1143844.1143846 ."},{"key":"5269_CR3","first-page":"227","volume-title":"Proceedings of the 39th annual ACM symposium on theory of computing (STOC \u201907)","author":"S. Arora","year":"2007","unstructured":"Arora, S., & Kale, S. (2007). A combinatorial, primal-dual approach to semidefinite programs. In Proceedings of the 39th annual ACM symposium on theory of computing (STOC \u201907) (pp. 227\u2013236). New York: ACM."},{"key":"5269_CR4","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1109\/SFCS.2005.35","volume-title":"46th annual symposium on foundations of computer science (FOCS \u201905)","author":"S. Arora","year":"2005","unstructured":"Arora, S., Hazan, E., & Kale, S. (2005). Fast algorithms for approximate semidefinite programming using the multiplicative weights update method. In 46th annual symposium on foundations of computer science (FOCS \u201905) (pp. 339\u2013348)."},{"key":"5269_CR5","volume-title":"Matrix mathematics: theory, facts, and formulas with application to linear systems theory","author":"D. S. Bernstein","year":"2005","unstructured":"Bernstein, D. S. (2005). Matrix mathematics: theory, facts, and formulas with application to linear systems theory. Princeton: Princeton University Press."},{"key":"5269_CR6","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0653-8","volume-title":"Matrix analysis","author":"R. Bhatia","year":"1997","unstructured":"Bhatia, R. (1997). Matrix analysis. Berlin: Springer."},{"key":"5269_CR7","first-page":"363","volume":"3","author":"O. Bousquet","year":"2002","unstructured":"Bousquet, O., & Warmuth, M. K. (2002). Tracking a small set of experts by mixing past posteriors. J. Mach. Learn. Res., 3, 363\u2013396.","journal-title":"J. Mach. Learn. Res."},{"key":"5269_CR8","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511804441","volume-title":"Convex optimization","author":"S. Boyd","year":"2004","unstructured":"Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge: Cambridge University Press."},{"key":"5269_CR9","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511546921","volume-title":"Prediction, learning, and games","author":"N. Cesa-Bianchi","year":"2006","unstructured":"Cesa-Bianchi, N., & Lugosi, G. (2006). Prediction, learning, and games. Cambridge: Cambridge University Press."},{"key":"5269_CR10","first-page":"217","volume-title":"Proceedings of the 18th annual conference on learning theory (COLT \u201905)","author":"N. Cesa-Bianchi","year":"2005","unstructured":"Cesa-Bianchi, N., Mansour, Y., & Stoltz, G. (2005). Improved second-order bounds for prediction with expert advice. In Proceedings of the 18th annual conference on learning theory (COLT \u201905) (pp. 217\u2013232). Berlin: Springer."},{"issue":"1","key":"5269_CR11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.1467-9965.1991.tb00002.x","volume":"1","author":"T. M. Cover","year":"1991","unstructured":"Cover, T. M. (1991). Universal portfolios. Math. Finance, 1(1), 1\u201329.","journal-title":"Math. Finance"},{"issue":"1","key":"5269_CR12","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1006\/jcss.1997.1504","volume":"55","author":"Y. Freund","year":"1997","unstructured":"Freund, Y., & Schapire, R.\u00a0E. (1997). A decision-theoretic generalization of on-line learning and an application to boosting. J. Comput. Syst. Sci., 55(1), 119\u2013139.","journal-title":"J. Comput. Syst. Sci."},{"key":"5269_CR13","first-page":"489","volume-title":"Proceedings of 20th annual conference on neural information processing systems (NIPS \u201906)","author":"G. J. Gordon","year":"2006","unstructured":"Gordon, G. J. (2006). No-regret algorithms for online convex programs. In B. Sch\u00f6lkopf, J. C. Platt, & T. Hoffman (Eds.), Proceedings of 20th annual conference on neural information processing systems (NIPS \u201906) (pp. 489\u2013496). Cambridge: MIT Press."},{"issue":"2\u20133","key":"5269_CR14","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/s10994-007-5016-8","volume":"69","author":"E. Hazan","year":"2007","unstructured":"Hazan, E., Agarwal, A., & Kale, S. (2007). Logarithmic regret algorithms for online convex optimization. Mach. Learn., 69(2\u20133), 169\u2013192.","journal-title":"Mach. Learn."},{"key":"5269_CR15","first-page":"314","volume-title":"Proceedings of the 23rd annual conference on learning theory (COLT \u201910)","author":"E. Hazan","year":"2010","unstructured":"Hazan, E., Kale, S., & Warmuth, M. K. (2010). On-line variance minimization in O(n 2) per trial? In Proceedings of the 23rd annual conference on learning theory (COLT \u201910) (pp. 314\u2013315)."},{"key":"5269_CR16","first-page":"469","volume-title":"Proceedings of the 20th annual conference on learning theory (COLT \u201907)","author":"D. Helmbold","year":"2007","unstructured":"Helmbold, D., & Warmuth, M. K. (2007). Learning permutations with exponential weights. In Proceedings of the 20th annual conference on learning theory (COLT \u201907) (pp. 469\u2013483). Berlin: Springer."},{"key":"5269_CR17","first-page":"1705","volume":"10","author":"D. Helmbold","year":"2009","unstructured":"Helmbold, D., & Warmuth, M. K. (2009). Learning permutations with exponential weights. J. Mach. Learn. Res., 10, 1705\u20131736.","journal-title":"J. Mach. Learn. Res."},{"issue":"1","key":"5269_CR18","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1023\/A:1007301011561","volume":"27","author":"D. Helmbold","year":"1997","unstructured":"Helmbold, D., Schapire, R.\u00a0E., Singer, Y., & Warmuth, M. K. (1997). A comparison of new and old algorithms for a mixture estimation problem. Mach. Learn., 27(1), 97\u2013119.","journal-title":"Mach. Learn."},{"issue":"4","key":"5269_CR19","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1111\/1467-9965.00058","volume":"8","author":"D. Helmbold","year":"1998","unstructured":"Helmbold, D., Schapire, R.\u00a0E., Singer, Y., & Warmuth, M. K. (1998). On-line portfolio selection using multiplicative updates. Math. Finance, 8(4), 325\u2013347.","journal-title":"Math. Finance"},{"issue":"6","key":"5269_CR20","doi-asserted-by":"crossref","first-page":"1291","DOI":"10.1109\/72.809075","volume":"10","author":"D. P. Helmbold","year":"1999","unstructured":"Helmbold, D. P., Kivinen, J., & Warmuth, M. K. (1999). Relative loss bounds for single neurons. IEEE Trans. Neural Netw., 10(6), 1291\u20131304.","journal-title":"IEEE Trans. Neural Netw."},{"issue":"2","key":"5269_CR21","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1023\/A:1007424614876","volume":"32","author":"M. Herbster","year":"1998","unstructured":"Herbster, M., & Warmuth, M. K. (1998). Tracking the best expert. Mach. Learn., 32(2), 151\u2013178.","journal-title":"Mach. Learn."},{"key":"5269_CR22","first-page":"281","volume":"1","author":"M. Herbster","year":"2001","unstructured":"Herbster, M., & Warmuth, M. K. (2001). Tracking the best linear predictor. J. Mach. Learn. Res., 1, 281\u2013309.","journal-title":"J. Mach. Learn. Res."},{"key":"5269_CR23","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1145\/1806689.1806768","volume-title":"Proceedings of the 42nd ACM symposium on theory of computing (STOC \u201910)","author":"R. Jain","year":"2010","unstructured":"Jain, R., Ji, Z., Upadhyay, S., & Watrous, J. (2010). QIP = PSPACE. In Proceedings of the 42nd ACM symposium on theory of computing (STOC \u201910) (pp. 573\u2013582)."},{"key":"5269_CR24","unstructured":"Kakade, S. M., Shalev-Shwartz, S., & Tewari, A. (2010). Regularizaton techniques for learning with matrices. arXiv:0910.0610v2 ."},{"key":"5269_CR25","unstructured":"Kalai, A. (2005). A perturbation that makes follow the leader? Equivalent to randomized weighted majority? Private communication."},{"issue":"3","key":"5269_CR26","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/j.jcss.2004.10.016","volume":"71","author":"A. Kalai","year":"2005","unstructured":"Kalai, A., & Vempala, S. (2005). Efficient algorithms for online decision problems. J. Comput. Syst. Sci., 71(3), 291\u2013307. doi: 10.1016\/j.jcss.2004.10.016 .","journal-title":"J. Comput. Syst. Sci."},{"issue":"1","key":"5269_CR27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/inco.1996.2612","volume":"132","author":"J. Kivinen","year":"1997","unstructured":"Kivinen, J., & Warmuth, M. K. (1997). Additive versus exponentiated gradient updates for linear prediction. Inf. Comput., 132(1), 1\u201364.","journal-title":"Inf. Comput."},{"key":"5269_CR28","series-title":"Lecture notes in artificial intelligence","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/3-540-49097-3_13","volume-title":"Computational learning theory, 4th European conference (EuroCOLT \u201999)","author":"J. Kivinen","year":"1999","unstructured":"Kivinen, J., & Warmuth, M. K. (1999). Averaging expert predictions. In Lecture notes in artificial intelligence: Vol. 1572. Computational learning theory, 4th European conference (EuroCOLT \u201999), Proceedings, Nordkirchen, Germany, March 29\u201331, 1999 (pp. 153\u2013167). Berlin: Springer."},{"issue":"3","key":"5269_CR29","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1023\/A:1017938623079","volume":"45","author":"J. Kivinen","year":"2001","unstructured":"Kivinen, J., & Warmuth, M. K. (2001). Relative loss bounds for multidimensional regression problems. Mach. Learn., 45(3), 301\u2013329.","journal-title":"Mach. Learn."},{"issue":"5","key":"5269_CR30","doi-asserted-by":"crossref","first-page":"1782","DOI":"10.1109\/TSP.2006.872551","volume":"54","author":"J. Kivinen","year":"2005","unstructured":"Kivinen, J., Warmuth, M. K., & Hassibi, B. (2005). The p-norm generalization of the LMS algorithm for adaptive filtering. IEEE Trans. Signal Process., 54(5), 1782\u20131793.","journal-title":"IEEE Trans. Signal Process."},{"key":"5269_CR31","first-page":"684","volume-title":"Proceedings of the 18th annual conference on learning theory (COLT \u201905)","author":"D. Kuzmin","year":"2005","unstructured":"Kuzmin, D., & Warmuth, M. K. (2005). Optimum follow the leader algorithm. In Proceedings of the 18th annual conference on learning theory (COLT \u201905) (pp. 684\u2013686). Berlin: Springer. Open problem."},{"key":"5269_CR32","series-title":"ACM international conference proceedings series","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1145\/1273496.1273555","volume-title":"Proceedings of the 24rd international conference on machine learning (ICML \u201907)","author":"D. Kuzmin","year":"2007","unstructured":"Kuzmin, D., & Warmuth, M. K. (2007). Online kernel PCA with entropic matrix updates. In ACM international conference proceedings series, Proceedings of the 24rd international conference on machine learning (ICML \u201907) (pp. 465\u2013471)."},{"issue":"4","key":"5269_CR33","first-page":"285","volume":"2","author":"N. Littlestone","year":"1988","unstructured":"Littlestone, N. (1988). Learning quickly when irrelevant attributes abound: a\u00a0new linear-threshold algorithm. Mach. Learn., 2(4), 285\u2013318.","journal-title":"Mach. Learn."},{"key":"5269_CR34","unstructured":"Littlestone, N. (1989). Mistake bounds and logarithmic linear-threshold learning algorithms. PhD thesis, Technical Report UCSC-CRL-89-11, University of California, Santa Cruz."},{"issue":"2","key":"5269_CR35","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1006\/inco.1994.1009","volume":"108","author":"N. Littlestone","year":"1994","unstructured":"Littlestone, N., & Warmuth, M. K. (1994). The weighted majority algorithm. Inf. Comput., 108(2), 212\u2013261. Preliminary version in Proceedings of the 30th annual symposium on foundations of computer science (FOCS \u201989).","journal-title":"Inf. Comput."},{"key":"5269_CR36","volume-title":"Quantum computation and quantum information","author":"M. A. Nielsen","year":"2000","unstructured":"Nielsen, M. A., & Chuang, I. L. (2000). Quantum computation and quantum information. Cambridge: Cambridge University Press."},{"key":"5269_CR37","first-page":"1265","volume-title":"Proceedings of 20th annual conference on neural information processing systems (NIPS \u201906)","author":"S. Shalev-Shwartz","year":"2006","unstructured":"Shalev-Shwartz, S., & Singer, Y. (2006). Convex repeated games and Fenchel duality. In B. Sch\u00f6lkopf, J. C. Platt, & T. Hoffman (Eds.), Proceedings of 20th annual conference on neural information processing systems (NIPS \u201906) (pp. 1265\u20131272). Cambridge: MIT Press."},{"key":"5269_CR38","first-page":"995","volume":"6","author":"K. Tsuda","year":"2005","unstructured":"Tsuda, K., R\u00e4tsch, G., & Warmuth, M. K. (2005). Matrix exponentiated gradient updates for on-line learning and Bregman projections. J. Mach. Learn. Res., 6, 995\u20131018.","journal-title":"J. Mach. Learn. Res."},{"key":"5269_CR39","first-page":"371","volume-title":"Proceedings of the 3rd annual workshop on computational learning theory","author":"V. Vovk","year":"1990","unstructured":"Vovk, V. (1990). Aggregating strategies. In Proceedings of the 3rd annual workshop on computational learning theory (pp. 371\u2013383). Morgan Kaufmann: San Mateo."},{"key":"5269_CR40","volume-title":"Proc. of the 20th annual conference on learning theory (COLT \u201907)","author":"M. K. Warmuth","year":"2007","unstructured":"Warmuth, M. K. (2007a). When is there a free matrix lunch. In Proc. of the 20th annual conference on learning theory (COLT \u201907). Berlin: Springer. Open problem."},{"key":"5269_CR41","volume-title":"Proceedings of the 24rd international conference on machine learning (ICML \u201907)","author":"M. K. Warmuth","year":"2007","unstructured":"Warmuth, M. K. (2007b). Winnowing subspaces. In Proceedings of the 24rd international conference on machine learning (ICML \u201907). New York: ACM."},{"key":"5269_CR42","first-page":"503","volume-title":"Proc. 22nd conference on uncertainty in artificial intelligence (UAI \u201906)","author":"M. K. Warmuth","year":"2006","unstructured":"Warmuth, M. K., & Kuzmin, D. (2006a). A Bayesian probability calculus for density matrices. In Proc. 22nd conference on uncertainty in artificial intelligence (UAI \u201906) (pp. 503\u2013511). Morgan Kaufmann: San Mateo. Journal submission: http:\/\/www.soe.ucsc.edu\/~manfred\/last\/bayescalc.pdf ."},{"key":"5269_CR43","first-page":"514","volume-title":"Proceedings of the 19th annual conference on learning theory (COLT \u201906)","author":"M. K. Warmuth","year":"2006","unstructured":"Warmuth, M. K., & Kuzmin, D. (2006b). Online variance minimization. In Proceedings of the 19th annual conference on learning theory (COLT \u201906) (pp. 514\u2013528). Berlin: Springer."},{"key":"5269_CR44","volume-title":"Advances in neural information processing systems 19 (NIPS \u201906)","author":"M. K. Warmuth","year":"2006","unstructured":"Warmuth, M. K., & Kuzmin, D. (2006c). Randomized PCA algorithms with regret bounds that are logarithmic in the dimension. In Advances in neural information processing systems 19 (NIPS \u201906). Cambridge: MIT Press."},{"key":"5269_CR45","first-page":"2217","volume":"9","author":"M. K. Warmuth","year":"2008","unstructured":"Warmuth, M. K., & Kuzmin, D. (2008). Randomized PCA algorithms with regret bounds that are logarithmic in the dimension. J. Mach. Learn. Res., 9, 2217\u20132250.","journal-title":"J. Mach. Learn. Res."},{"issue":"1\u20132","key":"5269_CR46","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s10994-009-5133-7","volume":"78","author":"M. K. Warmuth","year":"2010","unstructured":"Warmuth, M. K., & Kuzmin, D. (2010). Bayesian generalized probability calculus for density matrices. Mach. Learn., 78(1\u20132), 63\u2013101.","journal-title":"Mach. Learn."},{"key":"5269_CR47","first-page":"928","volume-title":"Proc. 20th int. conference on machine learning (ICML \u201903)","author":"M. Zinkevich","year":"2003","unstructured":"Zinkevich, M. (2003). Online convex programming and generalized infinitesimal gradient ascent. In Proc. 20th int. conference on machine learning (ICML \u201903) (pp. 928\u2013936)."}],"container-title":["Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-011-5269-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10994-011-5269-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-011-5269-0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,31]],"date-time":"2019-05-31T21:40:32Z","timestamp":1559338832000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10994-011-5269-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,11,20]]},"references-count":47,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,4]]}},"alternative-id":["5269"],"URL":"https:\/\/doi.org\/10.1007\/s10994-011-5269-0","relation":{},"ISSN":["0885-6125","1573-0565"],"issn-type":[{"value":"0885-6125","type":"print"},{"value":"1573-0565","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,11,20]]}}}