[{"article_processing_charge":"No","issue":"6","arxiv":1,"date_published":"2020-12-11T00:00:00Z","_id":"6748","abstract":[{"text":"Fitting a function by using linear combinations of a large number N of `simple' components is one of the most fruitful ideas in statistical learning. This idea lies at the core of a variety of methods, from two-layer neural networks to kernel regression, to boosting. In general, the resulting risk minimization problem is non-convex and is solved by gradient descent or its variants. Unfortunately, little is known about global convergence properties of these approaches.\r\nHere we consider the problem of learning a concave function f on a compact convex domain Ω⊆ℝd, using linear combinations of `bump-like' components (neurons). The parameters to be fitted are the centers of N bumps, and the resulting empirical risk minimization problem is highly non-convex. We prove that, in the limit in which the number of neurons diverges, the evolution of gradient descent converges to a Wasserstein gradient flow in the space of probability distributions over Ω. Further, when the bump width δ tends to 0, this gradient flow has a limit which is a viscous porous medium equation. Remarkably, the cost function optimized by this gradient flow exhibits a special property known as displacement convexity, which implies exponential convergence rates for N→∞, δ→0. Surprisingly, this asymptotic theory appears to capture well the behavior for moderate values of δ,N. Explaining this phenomenon, and understanding the dependence on δ,N in a quantitative manner remains an outstanding challenge.","lang":"eng"}],"oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1901.01375","open_access":"1"}],"publication_status":"published","volume":48,"oa_version":"Preprint","year":"2020","article_type":"original","publication_identifier":{"eissn":["1941-7330"],"issn":["1932-6157"]},"external_id":{"arxiv":["1901.01375"],"isi":["000598369200021"]},"date_updated":"2024-03-06T08:28:50Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"3619-3642","date_created":"2019-07-31T09:39:42Z","month":"12","isi":1,"publisher":"Institute of Mathematical Statistics","status":"public","intvolume":"        48","publication":"Annals of Statistics","quality_controlled":"1","department":[{"_id":"MaMo"}],"title":"Analysis of a two-layer neural network via displacement convexity","citation":{"short":"A. Javanmard, M. Mondelli, A. Montanari, Annals of Statistics 48 (2020) 3619–3642.","apa":"Javanmard, A., Mondelli, M., &#38; Montanari, A. (2020). Analysis of a two-layer neural network via displacement convexity. <i>Annals of Statistics</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/20-AOS1945\">https://doi.org/10.1214/20-AOS1945</a>","ama":"Javanmard A, Mondelli M, Montanari A. Analysis of a two-layer neural network via displacement convexity. <i>Annals of Statistics</i>. 2020;48(6):3619-3642. doi:<a href=\"https://doi.org/10.1214/20-AOS1945\">10.1214/20-AOS1945</a>","ista":"Javanmard A, Mondelli M, Montanari A. 2020. Analysis of a two-layer neural network via displacement convexity. Annals of Statistics. 48(6), 3619–3642.","ieee":"A. Javanmard, M. Mondelli, and A. Montanari, “Analysis of a two-layer neural network via displacement convexity,” <i>Annals of Statistics</i>, vol. 48, no. 6. Institute of Mathematical Statistics, pp. 3619–3642, 2020.","chicago":"Javanmard, Adel, Marco Mondelli, and Andrea Montanari. “Analysis of a Two-Layer Neural Network via Displacement Convexity.” <i>Annals of Statistics</i>. Institute of Mathematical Statistics, 2020. <a href=\"https://doi.org/10.1214/20-AOS1945\">https://doi.org/10.1214/20-AOS1945</a>.","mla":"Javanmard, Adel, et al. “Analysis of a Two-Layer Neural Network via Displacement Convexity.” <i>Annals of Statistics</i>, vol. 48, no. 6, Institute of Mathematical Statistics, 2020, pp. 3619–42, doi:<a href=\"https://doi.org/10.1214/20-AOS1945\">10.1214/20-AOS1945</a>."},"author":[{"last_name":"Javanmard","first_name":"Adel","full_name":"Javanmard, Adel"},{"orcid":"0000-0002-3242-7020","id":"27EB676C-8706-11E9-9510-7717E6697425","last_name":"Mondelli","first_name":"Marco","full_name":"Mondelli, Marco"},{"first_name":"Andrea","full_name":"Montanari, Andrea","last_name":"Montanari"}],"type":"journal_article","day":"11","doi":"10.1214/20-AOS1945","language":[{"iso":"eng"}]}]