[{"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1901.01375"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","type":"journal_article","date_published":"2020-12-11T00:00:00Z","publication_identifier":{"issn":["1932-6157"],"eissn":["1941-7330"]},"oa":1,"language":[{"iso":"eng"}],"publication":"Annals of Statistics","oa_version":"Preprint","month":"12","volume":48,"year":"2020","citation":{"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>.","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.","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.","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>.","short":"A. Javanmard, M. Mondelli, A. Montanari, Annals of Statistics 48 (2020) 3619–3642."},"date_updated":"2024-03-06T08:28:50Z","external_id":{"arxiv":["1901.01375"],"isi":["000598369200021"]},"isi":1,"day":"11","arxiv":1,"doi":"10.1214/20-AOS1945","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"}],"quality_controlled":"1","page":"3619-3642","publisher":"Institute of Mathematical Statistics","article_type":"original","_id":"6748","issue":"6","author":[{"first_name":"Adel","last_name":"Javanmard","full_name":"Javanmard, Adel"},{"orcid":"0000-0002-3242-7020","full_name":"Mondelli, Marco","first_name":"Marco","last_name":"Mondelli","id":"27EB676C-8706-11E9-9510-7717E6697425"},{"last_name":"Montanari","first_name":"Andrea","full_name":"Montanari, Andrea"}],"department":[{"_id":"MaMo"}],"date_created":"2019-07-31T09:39:42Z","article_processing_charge":"No","publication_status":"published","intvolume":"        48","title":"Analysis of a two-layer neural network via displacement convexity"}]