[{"oa_version":"Preprint","_id":"6748","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":" 48","status":"public","issue":"6","abstract":[{"lang":"eng"}],"type":"journal_article","date_published":"2020-12-11T00:00:00Z","citation":{"chicago":"Javanmard, Adel, Marco Mondelli, and Andrea Montanari. “Analysis of a Two-Layer Neural Network via Displacement Convexity.” Annals of Statistics. Institute of Mathematical Statistics, 2020. https://doi.org/10.1214/20-AOS1945.","short":"A. Javanmard, M. Mondelli, A. Montanari, Annals of Statistics 48 (2020) 3619–3642.","mla":"Javanmard, Adel, et al. “Analysis of a Two-Layer Neural Network via Displacement Convexity.” Annals of Statistics, vol. 48, no. 6, Institute of Mathematical Statistics, 2020, pp. 3619–42, doi:10.1214/20-AOS1945.","apa":"Javanmard, A., Mondelli, M., & Montanari, A. (2020). Analysis of a two-layer neural network via displacement convexity. Annals of Statistics. Institute of Mathematical Statistics. https://doi.org/10.1214/20-AOS1945","ieee":"A. Javanmard, M. Mondelli, and A. Montanari, “Analysis of a two-layer neural network via displacement convexity,” Annals of Statistics, vol. 48, no. 6. Institute of Mathematical Statistics, pp. 3619–3642, 2020.","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."},"publication":"Annals of Statistics","page":"3619-3642","article_type":"original","uri_base":"https://research-explorer.ista.ac.at","article_processing_charge":"No","day":"11","dc":{"type":["info:eu-repo/semantics/article","doc-type:article","text","http://purl.org/coar/resource_type/c_6501"],"identifier":["https://research-explorer.ista.ac.at/record/6748"],"description":["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."],"creator":["Javanmard, Adel","Mondelli, Marco","Montanari, Andrea"],"rights":["info:eu-repo/semantics/openAccess"],"source":["Javanmard A, Mondelli M, Montanari A. Analysis of a two-layer neural network via displacement convexity. Annals of Statistics. 2020;48(6):3619-3642. doi:10.1214/20-AOS1945"],"title":["Analysis of a two-layer neural network via displacement convexity"],"relation":["info:eu-repo/semantics/altIdentifier/doi/10.1214/20-AOS1945","info:eu-repo/semantics/altIdentifier/issn/1932-6157","info:eu-repo/semantics/altIdentifier/issn/1941-7330","info:eu-repo/semantics/altIdentifier/wos/000598369200021","info:eu-repo/semantics/altIdentifier/arxiv/1901.01375"],"publisher":["Institute of Mathematical Statistics"],"date":["2020"],"language":["eng"]},"author":[{"last_name":"Javanmard","first_name":"Adel"},{"orcid":"0000-0002-3242-7020","id":"27EB676C-8706-11E9-9510-7717E6697425","last_name":"Mondelli","first_name":"Marco"},{"first_name":"Andrea","last_name":"Montanari"}],"volume":48,"date_updated":"2024-03-06T08:28:50Z","dini_type":"doc-type:article","date_created":"2019-07-31T09:39:42Z","department":[{"_id":"MaMo","tree":[{"_id":"ResearchGroups"},{"_id":"IST"}]}],"publication_status":"published","creator":{"id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","login":"apreinsp"},"language":[{}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1901.01375"}],"external_id":{"arxiv":[],"isi":[]},"oa":1,"isi":1,"quality_controlled":"1","publication_identifier":{"eissn":[],"issn":[]},"month":"12"}]