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A rigorous framework for the mean field limit of multilayer neural networks 多层神经网络平均场极限的严格框架
Mathematical statistics and learning Pub Date : 2023-10-09 DOI: 10.4171/msl/42
Phan-Minh Nguyen, Huy Tuan Pham
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引用次数: 70
U-statistics of growing order and sub-Gaussian mean estimators with sharp constants 增长阶的u统计量和具有尖锐常数的亚高斯均值估计量
Mathematical statistics and learning Pub Date : 2023-10-09 DOI: 10.4171/msl/43
Stanislav Minsker
{"title":"U-statistics of growing order and sub-Gaussian mean estimators with sharp constants","authors":"Stanislav Minsker","doi":"10.4171/msl/43","DOIUrl":"https://doi.org/10.4171/msl/43","url":null,"abstract":"This paper addresses the following question: given a sample of i.i.d. random variables with finite variance, can one construct an estimator of the unknown mean that performs nearly as well as if the data were normally distributed? One of the most popular examples achieving this goal is the median of means estimator. However, it is inefficient in a sense that the constants in the resulting bounds are suboptimal. We show that a permutation-invariant modification of the median of means estimator admits deviation guarantees that are sharp up to $1+o(1)$ factor if the underlying distribution possesses more than $frac{3+sqrt{5}}{2}approx 2.62$ moments and is absolutely continuous with respect to the Lebesgue measure. This result yields potential improvements for a variety of algorithms that rely on the median of means estimator as a building block. At the core of our argument is are the new deviation inequalities for the U-statistics of order that is allowed to grow with the sample size, a result that could be of independent interest.","PeriodicalId":489458,"journal":{"name":"Mathematical statistics and learning","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135142251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
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