--- res: bibo_abstract: - '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.@eng' bibo_authorlist: - foaf_Person: foaf_givenName: Aleksandr foaf_name: Shevchenko, Aleksandr foaf_surname: Shevchenko foaf_workInfoHomepage: http://www.librecat.org/personId=F2B06EC2-C99E-11E9-89F0-752EE6697425 - foaf_Person: foaf_givenName: Vyacheslav foaf_name: Kungurtsev, Vyacheslav foaf_surname: Kungurtsev - foaf_Person: foaf_givenName: Marco foaf_name: Mondelli, Marco foaf_surname: Mondelli foaf_workInfoHomepage: http://www.librecat.org/personId=27EB676C-8706-11E9-9510-7717E6697425 orcid: 0000-0002-3242-7020 bibo_issue: '130' bibo_volume: 23 dct_date: 2022^xs_gYear dct_isPartOf: - http://id.crossref.org/issn/1532-4435 - http://id.crossref.org/issn/1533-7928 dct_language: eng dct_publisher: Journal of Machine Learning Research@ dct_title: Mean-field analysis of piecewise linear solutions for wide ReLU networks@ ...