article published yes Aleksandr Shevchenko author F2B06EC2-C99E-11E9-89F0-752EE6697425 Vyacheslav Kungurtsev author Marco Mondelli author 27EB676C-8706-11E9-9510-7717E66974250000-0002-3242-7020 MaMo department DaAl department Prix Lopez-Loretta 2019 - Marco Mondelli project 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. https://research-explorer.ista.ac.at/download/11420/11422/21-1365.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ Journal of Machine Learning Research2022 eng 1532-4435 1533-7928 2111.02278 231301-55 https://research-explorer.ista.ac.at/record/17465 https://www.jmlr.org/papers/v23/21-1365.html A. Shevchenko, V. Kungurtsev, M. Mondelli, Journal of Machine Learning Research 23 (2022) 1–55. 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. Shevchenko, Alexander, 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. 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. 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. Shevchenko, Alexander, 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. 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. 114202022-05-29T22:01:54Z2026-07-11T22:30:48Z