{"publication_identifier":{"eissn":["1533-7928"],"issn":["1532-4435"]},"external_id":{"arxiv":["2111.02278"]},"author":[{"first_name":"Aleksandr","last_name":"Shevchenko","full_name":"Shevchenko, Aleksandr","id":"F2B06EC2-C99E-11E9-89F0-752EE6697425"},{"full_name":"Kungurtsev, Vyacheslav","last_name":"Kungurtsev","first_name":"Vyacheslav"},{"orcid":"0000-0002-3242-7020","first_name":"Marco","last_name":"Mondelli","id":"27EB676C-8706-11E9-9510-7717E6697425","full_name":"Mondelli, Marco"}],"publication":"Journal of Machine Learning Research","has_accepted_license":"1","page":"1-55","_id":"11420","language":[{"iso":"eng"}],"citation":{"ieee":"A. Shevchenko, V. Kungurtsev, and M. Mondelli, “Mean-field analysis of piecewise linear solutions for wide ReLU networks,” <i>Journal of Machine Learning Research</i>, vol. 23, no. 130. Journal of Machine Learning Research, pp. 1–55, 2022.","mla":"Shevchenko, Aleksandr, et al. “Mean-Field Analysis of Piecewise Linear Solutions for Wide ReLU Networks.” <i>Journal of Machine Learning Research</i>, vol. 23, no. 130, Journal of Machine Learning Research, 2022, pp. 1–55.","short":"A. Shevchenko, V. Kungurtsev, M. Mondelli, Journal of Machine Learning Research 23 (2022) 1–55.","ama":"Shevchenko A, Kungurtsev V, Mondelli M. Mean-field analysis of piecewise linear solutions for wide ReLU networks. <i>Journal of Machine Learning Research</i>. 2022;23(130):1-55.","chicago":"Shevchenko, Aleksandr, Vyacheslav Kungurtsev, and Marco Mondelli. “Mean-Field Analysis of Piecewise Linear Solutions for Wide ReLU Networks.” <i>Journal of Machine Learning Research</i>. Journal of Machine Learning Research, 2022.","apa":"Shevchenko, A., Kungurtsev, V., &#38; Mondelli, M. (2022). Mean-field analysis of piecewise linear solutions for wide ReLU networks. <i>Journal of Machine Learning Research</i>. Journal of Machine Learning Research.","ista":"Shevchenko A, Kungurtsev V, Mondelli M. 2022. Mean-field analysis of piecewise linear solutions for wide ReLU networks. Journal of Machine Learning Research. 23(130), 1–55."},"article_type":"original","year":"2022","volume":23,"intvolume":"        23","project":[{"_id":"059876FA-7A3F-11EA-A408-12923DDC885E","name":"Prix Lopez-Loretta 2019 - Marco Mondelli"}],"ddc":["000"],"acknowledgement":"We would like to thank Mert Pilanci for several exploratory discussions in the early stage\r\nof the project, Jan Maas for clarifications about Jordan et al. (1998), and Max Zimmer for\r\nsuggestive numerical experiments. A. Shevchenko and M. Mondelli are partially supported\r\nby the 2019 Lopez-Loreta Prize. V. Kungurtsev acknowledges support to the OP VVV\r\nproject CZ.02.1.01/0.0/0.0/16 019/0000765 Research Center for Informatics.\r\n","file_date_updated":"2022-05-30T08:22:55Z","date_updated":"2024-09-10T13:03:17Z","tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"month":"04","article_processing_charge":"No","day":"01","type":"journal_article","file":[{"file_name":"21-1365.pdf","creator":"cchlebak","access_level":"open_access","content_type":"application/pdf","relation":"main_file","checksum":"d4ff5d1affb34848b5c5e4002483fc62","date_created":"2022-05-30T08:22:55Z","file_size":1521701,"file_id":"11422","success":1,"date_updated":"2022-05-30T08:22:55Z"}],"abstract":[{"lang":"eng","text":"Understanding the properties of neural networks trained via stochastic gradient descent (SGD) is at the heart of the theory of deep learning. In this work, we take a mean-field view, and consider a two-layer ReLU network trained via noisy-SGD for a univariate regularized regression problem. Our main result is that SGD with vanishingly small noise injected in the gradients is biased towards a simple solution: at convergence, the ReLU network implements a piecewise linear map of the inputs, and the number of “knot” points -- i.e., points where the tangent of the ReLU network estimator changes -- between two consecutive training inputs is at most three. In particular, as the number of neurons of the network grows, the SGD dynamics is captured by the solution of a gradient flow and, at convergence, the distribution of the weights approaches the unique minimizer of a related free energy, which has a Gibbs form. Our key technical contribution consists in the analysis of the estimator resulting from this minimizer: we show that its second derivative vanishes everywhere, except at some specific locations which represent the “knot” points. We also provide empirical evidence that knots at locations distinct from the data points might occur, as predicted by our theory."}],"department":[{"_id":"MaMo"},{"_id":"DaAl"}],"issue":"130","publication_status":"published","date_published":"2022-04-01T00:00:00Z","scopus_import":"1","publisher":"Journal of Machine Learning Research","date_created":"2022-05-29T22:01:54Z","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","oa_version":"Published Version","oa":1,"status":"public","related_material":{"link":[{"relation":"other","url":"https://www.jmlr.org/papers/v23/21-1365.html"}]},"quality_controlled":"1","title":"Mean-field analysis of piecewise linear solutions for wide ReLU networks"}