{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T02:06:37Z","timestamp":1775527597107,"version":"3.50.1"},"reference-count":26,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T00:00:00Z","timestamp":1759276800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>We prove large and moderate deviations for the output of Gaussian fully connected neural networks. The main achievements concern deep neural networks (i.e. when the model has more than one hidden layer) and hold for bounded and continuous pre-activation functions. However, for deep neural networks fed by a single input, we have results even if the pre-activation is ReLU. When the network is shallow (i.e. there is exactly one hidden layer), the large and moderate principles hold for quite general pre-activation functions.<\/jats:p>","DOI":"10.1017\/jpr.2025.10024","type":"journal-article","created":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T07:22:00Z","timestamp":1759303320000},"page":"96-115","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["Large and moderate deviations for Gaussian neural networks"],"prefix":"10.1017","volume":"63","author":[{"given":"Claudio","family":"Macci","sequence":"first","affiliation":[{"name":"Universit\u00e0 di Roma Tor Vergata"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Barbara","family":"Pacchiarotti","sequence":"additional","affiliation":[{"name":"Universit\u00e0 di Roma Tor Vergata"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Giovanni Luca","family":"Torrisi","sequence":"additional","affiliation":[{"name":"Consiglio Nazionale delle Ricerche"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2025,10,1]]},"reference":[{"key":"S0021900225100247_ref6","doi-asserted-by":"publisher","DOI":"10.3150\/23-BEJ1605"},{"key":"S0021900225100247_ref15","doi-asserted-by":"publisher","DOI":"10.1214\/23-AAP1933"},{"key":"S0021900225100247_ref14","first-page":"2672","article-title":"Generative adversarial nets","volume":"27","author":"Goodfellow","year":"2014","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"S0021900225100247_ref1","doi-asserted-by":"publisher","DOI":"10.1287\/stsy.2023.0033"},{"key":"S0021900225100247_ref19","doi-asserted-by":"publisher","DOI":"10.1038\/nature14539"},{"key":"S0021900225100247_ref10","unstructured":"[10] Eldan, R. , Mikulincer, D. and Schramm, T. (2021). Non-asymptotic approximations of neural networks by Gaussian processes. Proc. Mach. Learn. Res. 134, 1754\u20131775."},{"key":"S0021900225100247_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-025-01360-1"},{"key":"S0021900225100247_ref11","doi-asserted-by":"publisher","DOI":"10.3150\/22-BEJ1553"},{"key":"S0021900225100247_ref7","doi-asserted-by":"publisher","DOI":"10.15559\/23-VMSTA238"},{"key":"S0021900225100247_ref21","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0745-0_2"},{"key":"S0021900225100247_ref22","doi-asserted-by":"publisher","DOI":"10.1017\/9781009023405"},{"key":"S0021900225100247_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2024.101668"},{"key":"S0021900225100247_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s10994-024-06578-z"},{"key":"S0021900225100247_ref13","doi-asserted-by":"publisher","DOI":"10.1017\/jpr.2023.118"},{"key":"S0021900225100247_ref23","doi-asserted-by":"crossref","unstructured":"[23] Saulis, L. and Statulevi\u010dius, V. A. (1991). Limit Theorems for Large Deviations (Math. Appl. (Sov. Ser.) 73). Kluwer, Dordrecht.","DOI":"10.1007\/978-94-011-3530-6"},{"key":"S0021900225100247_ref16","first-page":"1","article-title":"Random fully connected neural networks as perturbatively solvable hierarchies","volume":"25","author":"Hanin","year":"2024","journal-title":"J. Mach. Learn. Res."},{"key":"S0021900225100247_ref18","doi-asserted-by":"publisher","DOI":"10.1017\/apr.2023.3"},{"key":"S0021900225100247_ref24","doi-asserted-by":"publisher","DOI":"10.1137\/18M1192184"},{"key":"S0021900225100247_ref5","first-page":"45","article-title":"Non-asymptotic approximations of Gaussian neural networks via second-order Poincar\u00e9 inequalities","volume":"253","author":"Bordino","year":"2024","journal-title":"Prog. Mach. Learn. Res."},{"key":"S0021900225100247_ref17","doi-asserted-by":"publisher","DOI":"10.1142\/S0219493725500029"},{"key":"S0021900225100247_ref26","unstructured":"[26] Zavatone-Veth, J. A. and Pehlevan, C. (2021). Exact marginal prior distributions of finite Bayesian neural networks. In Advances in Neural Information Processing Systems 34, eds. M. Ranzato, A. Beygelzimer, Y. Dauphin, P. S. Liang and J. Wortman Vaughan. Curran Associates, Red Hook, NY, pp. 3364\u20133375."},{"key":"S0021900225100247_ref20","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8121\/ab6a6f"},{"key":"S0021900225100247_ref8","first-page":"147","article-title":"Large deviations for joint distributions and statistical applications","volume":"59","author":"Chaganty","year":"1997","journal-title":"Sankhy\u0101 A"},{"key":"S0021900225100247_ref9","doi-asserted-by":"crossref","unstructured":"[9] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications (Stoch. Model. Appl. Prob. 38). Springer, Berlin.","DOI":"10.1007\/978-3-642-03311-7"},{"key":"S0021900225100247_ref4","unstructured":"[4] Bathia, R. (1997). Matrix Analysis (Grad. Texts Math. 169). Springer, Berlin."},{"key":"S0021900225100247_ref25","unstructured":"[25] Vogel, Q. (2024). Large deviations of Gaussian neural networks with ReLU activation. Preprint, arXiv:2405.16958."}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900225100247","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T01:25:35Z","timestamp":1775525135000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900225100247\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,1]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0021900225100247"],"URL":"https:\/\/doi.org\/10.1017\/jpr.2025.10024","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,1]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}