{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:34:30Z","timestamp":1777448070317,"version":"3.51.4"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"2-3","license":[{"start":{"date-parts":[[1996,11,1]],"date-time":"1996-11-01T00:00:00Z","timestamp":846806400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[1996,11,1]],"date-time":"1996-11-01T00:00:00Z","timestamp":846806400000},"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":["Machine Learning"],"published-print":{"date-parts":[[1996,11]]},"DOI":"10.1023\/a:1026499208981","type":"journal-article","created":{"date-parts":[[2003,11,6]],"date-time":"2003-11-06T11:45:40Z","timestamp":1068119140000},"page":"195-236","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Rigorous Learning Curve Bounds from Statistical Mechanics"],"prefix":"10.1007","volume":"25","author":[{"given":"David","family":"Haussler","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Kearns","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"H. Sebastian","family":"Seung","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Naftali","family":"Tishby","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","reference":[{"issue":"4","key":"118345_CR1","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1162\/neco.1992.4.4.605","volume":"4","author":"S. Amari","year":"1992","unstructured":"Amari, S., Fujita, N., & Shinomoto, S. (1992). Four types of learning curves. Neural Computation, 4(4):605\u2013618.","journal-title":"Neural Computation"},{"key":"118345_CR2","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1162\/neco.1991.3.3.386","volume":"3","author":"E.B. Baum","year":"1991","unstructured":"Baum, E.B., & Lyuu, Y.-D. (1991). The transition to perfect generalization in perceptrons. Neural Comput., 3:386\u2013401.","journal-title":"Neural Comput."},{"issue":"2","key":"118345_CR3","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/0304-3975(91)90026-X","volume":"86","author":"G. Benedek","year":"1991","unstructured":"Benedek, G., & Itai, A. (1991). Learnability with respect to fixed distributions. Theoret. Comput. Sci., 86(2):377\u2013389.","journal-title":"Theoret. Comput. Sci."},{"key":"118345_CR4","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1162\/neco.1992.4.2.249","volume":"4","author":"D. Cohn","year":"1992","unstructured":"Cohn, D., & Tesauro, G. (1992). How tight are the Vapnik-Chervonenkis bounds? Neural Comput., 4:249\u2013269.","journal-title":"Neural Comput."},{"key":"118345_CR5","unstructured":"Cover, T., & Thomas, J. (1991). Elements of Information Theory, Wiley."},{"key":"118345_CR6","doi-asserted-by":"crossref","unstructured":"Devroye, L., & Lugosi, G. (1994). Lower bounds in pattern recognition and learning. Preprint.","DOI":"10.1016\/0031-3203(94)00141-8"},{"issue":"6","key":"118345_CR7","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1214\/aop\/1176995384","volume":"6","author":"R.M. Dudley","year":"1978","unstructured":"Dudley, R.M. (1978). Central limit theorems for empirical measures. Annals of Probability, 6(6):899\u2013929.","journal-title":"Annals of Probability"},{"issue":"3","key":"118345_CR8","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/0890-5401(89)90002-3","volume":"82","author":"A. Ehrenfeucht","year":"1989","unstructured":"Ehrenfeucht, A., Haussler, D., Kearns, M., & Valiant, L. (1989). Ageneral lower bound on the number of examples needed for learning. Information and Computation, 82(3):247\u2013251.","journal-title":"Information and Computation"},{"key":"118345_CR9","first-page":"6893","volume":"26","author":"A. Engel","year":"1993","unstructured":"Engel, A., & Fink, W. (1993). Statistical mechanics calculation of Vapnik Chervonenkis bounds for perceptrons. J. Phys., 26:6893\u20136914.","journal-title":"J. Phys."},{"key":"118345_CR10","doi-asserted-by":"crossref","first-page":"1772","DOI":"10.1103\/PhysRevLett.71.1772","volume":"71","author":"A. Engel","year":"1993","unstructured":"Engel, A., & van den Broeck, C. (1993). Systems that can learn from examples: replica calculation of uniform convergence bounds for the perceptron. Phys. Rev. Lett., 71:1772\u20131775.","journal-title":"Phys. Rev. Lett."},{"key":"118345_CR11","first-page":"257","volume":"A21","author":"E. Gardner","year":"1988","unstructured":"Gardner, E. (1988). The space of interactions in neural network models. J. Phys., A21:257\u2013270.","journal-title":"J. Phys."},{"key":"118345_CR12","first-page":"1983","volume":"A22","author":"E. Gardner","year":"1989","unstructured":"Gardner, E., & Derrida, B. (1989). Three unfinished works on the optimal storage capacity of networks. J. Phys., A22:1983\u20131994.","journal-title":"J. Phys."},{"key":"118345_CR13","first-page":"217","volume-title":"Proceedings of the 3rd Workshop on Computational Learning Theory","author":"S.A. Goldman","year":"1990","unstructured":"Goldman, S.A., Kearns, M.J., & Schapire, R.E. (1990). On the sample complexity of weak learning. In Proceedings of the 3rd Workshop on Computational Learning Theory(pp. 217\u2013231), San Mateo, CA: Morgan Kaufmann."},{"key":"118345_CR14","doi-asserted-by":"crossref","first-page":"7097","DOI":"10.1103\/PhysRevA.41.7097","volume":"A41","author":"G. Gy\u00f6rgyi","year":"1990","unstructured":"Gy\u00f6rgyi, G. (1990). First-order transition to perfect generalization in a neural network with binary synapses. Phys. Rev., A41:7097\u20137100.","journal-title":"Phys. Rev."},{"key":"118345_CR15","unstructured":"Gyorgyi, G., & Tishby, N. (1990). Statistical theory of learning a rule. In K. Thuemann & R. Koeberle (Eds.), Neural Networks and Spin Glasses, World Scientific."},{"issue":"1","key":"118345_CR16","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/0890-5401(92)90010-D","volume":"100","author":"D. Haussler","year":"1992","unstructured":"Haussler, D. (1992). Decision-theoretic generalizations of the PAC model for neural net and other learning applications. Information and Computation, 100(1):78\u2013150.","journal-title":"Information and Computation"},{"key":"118345_CR17","first-page":"61","volume-title":"Proceedings of the 4th Workshop on Computational Learning Theory","author":"D. Haussler","year":"1991","unstructured":"Haussler, D., Kearns, M., & Schapire, R.E. (1991). Bounds on the sample complexity of Bayesian learning using information theory and the VC dimension. In Proceedings of the 4th Workshop on Computational Learning Theory (pp. 61\u201374), San Mateo, CA: Morgan Kaufmann."},{"key":"118345_CR18","doi-asserted-by":"crossref","unstructured":"Levin, E., Tishby, N., & Solla, S. (1989). A statistical approach to learning and generalization in neural networks. In R. Rivest, (Ed.), Proc. 3rd Annu. Workshop on Comput. Learning Theory, Morgan Kaufmann.","DOI":"10.1016\/B978-0-08-094829-4.50020-9"},{"key":"118345_CR19","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1162\/neco.1992.4.6.854","volume":"4","author":"Y.-D. Lyuu","year":"1992","unstructured":"Lyuu, Y.-D., & Rivin, I. (1992) Tight bounds on transition to perfect generalization in perceptrons. Neural Comput., 4:854\u2013862.","journal-title":"Neural Comput."},{"key":"118345_CR20","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1162\/neco.1991.3.2.258","volume":"3","author":"G.L. Martin","year":"1991","unstructured":"Martin, G.L., & Pittman, J.A. (1991). Recognizing hand-printed letters and digits using backpropagation learning. Neural Comput., 3:258\u2013267.","journal-title":"Neural Comput."},{"issue":"1","key":"118345_CR21","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1023\/A:1022634402267","volume":"8","author":"E. Oblow","year":"1992","unstructured":"Oblow, E. (1992). Implementing Valiant's learnability theory using random sets. Machine Learning, 8(1):45\u201374.","journal-title":"Machine Learning"},{"key":"118345_CR22","doi-asserted-by":"crossref","unstructured":"Pollard, D. (1984). Convergence of Stochastic Processes, Springer-Verlag.","DOI":"10.1007\/978-1-4612-5254-2"},{"issue":"4","key":"118345_CR23","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1023\/A:1022601518391","volume":"9","author":"W. Sarrett","year":"1992","unstructured":"Sarrett, W., & Pazzani, M. (1992). Average case analysis of empirical and explanation-based learning algorithms. Machine Learning, 9(4):349\u2013372.","journal-title":"Machine Learning"},{"key":"118345_CR24","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1162\/neco.1990.2.3.374","volume":"2","author":"D.B. Schwartz","year":"1990","unstructured":"Schwartz, D.B., Samalam, V.K., Denker, J.S., & Solla, S.A. (1990). Exhaustive learning. Neural Comput., 2:374\u2013385.","journal-title":"Neural Comput."},{"key":"118345_CR25","doi-asserted-by":"crossref","first-page":"6056","DOI":"10.1103\/PhysRevA.45.6056","volume":"A45","author":"H.S. Seung","year":"1992","unstructured":"Seung, H.S., Sompolinsky, H., & Tishby, N. (1992). Statistical mechanics of learning from examples. Physical Review, A45:6056\u20136091.","journal-title":"Physical Review"},{"key":"118345_CR26","first-page":"402","volume-title":"Proceedings of the 6th Annual ACM Conference on Computational Learning Theory","author":"H.U. Simon","year":"1993","unstructured":"Simon, H.U. (1993). General bounds on the number of examples needed for learning probabilistic concepts. In Proceedings of the 6th Annual ACM Conference on Computational Learning Theory(pp. 402\u2013411), New York, NY: ACM Press."},{"key":"118345_CR27","first-page":"112","volume-title":"Proc. 4th Annu. Workshop on Comput. Learning Theory","author":"H. Sompolinsky","year":"1991","unstructured":"Sompolinsky, H., Seung, H.S., & Tishby, N. (1991). Learning curves in large neural networks. In Proc. 4th Annu. Workshop on Comput. Learning Theory (pp. 112\u2013127), San Mateo, CA: Morgan Kaufmann."},{"issue":"13","key":"118345_CR28","doi-asserted-by":"crossref","first-page":"1683","DOI":"10.1103\/PhysRevLett.65.1683","volume":"65","author":"H. Sompolinsky","year":"1990","unstructured":"Sompolinsky, H., Tishby, N., & Seung, H.S. (1990). Learning from examples in large neural networks. Phys. Rev. Lett., 65(13):1683\u20131686.","journal-title":"Phys. Rev. Lett."},{"issue":"5","key":"118345_CR29","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1162\/neco.1994.6.5.851","volume":"6","author":"V. Vapnik","year":"1994","unstructured":"Vapnik, V., Levin, E., & LeCun, Y. (1994). Measuring the VC dimension of a learning machine. Neural Compu-tation, 6(5):851\u2013876.","journal-title":"Neural Compu-tation"},{"key":"118345_CR30","volume-title":"Estimation of Dependences Based on Empirical Data","author":"V.N. Vapnik","year":"1982","unstructured":"Vapnik, V.N. (1982). Estimation of Dependences Based on Empirical Data, Springer-Verlag, New York."},{"issue":"2","key":"118345_CR31","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1137\/1116025","volume":"16","author":"V.N. Vapnik","year":"1971","unstructured":"Vapnik, V.N., & Chervonenkis, A.Y. (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16(2):264\u2013280.","journal-title":"Theory of Probability and its Applications"},{"key":"118345_CR32","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1103\/RevModPhys.65.499","volume":"65","author":"T.L.H. Watkin","year":"1993","unstructured":"Watkin, T.L.H., Rau, A., & Biehl, M. (1993). The statistical mechanics of learning a rule. Rev. Mod. Phys., 65:499\u2013556.","journal-title":"Rev. Mod. Phys."}],"container-title":["Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1023\/A:1026499208981.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1023\/A:1026499208981\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1023\/A:1026499208981.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T11:34:37Z","timestamp":1752147277000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1023\/A:1026499208981"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,11]]},"references-count":32,"journal-issue":{"issue":"2-3","published-print":{"date-parts":[[1996,11]]}},"alternative-id":["118345"],"URL":"https:\/\/doi.org\/10.1023\/a:1026499208981","relation":{},"ISSN":["0885-6125","1573-0565"],"issn-type":[{"value":"0885-6125","type":"print"},{"value":"1573-0565","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,11]]},"assertion":[{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}