
Open access
Date
2017-08Type
- Journal Article
Abstract
Rates of convergence for empirical risk minimizers have been well studied in the literature. In this paper, we aim to provide a complementary set of results, in particular by showing that after normalization, the risk of the empirical minimizer concentrates on a single point. Such results have been established by Chatterjee (The Annals of Statistics, 42(6):2340–2381 2014) for constrained estimators in the normal sequence model. We first generalize and sharpen this result to regularized least squares with convex penalties, making use of a “direct” argument based on Borell’s theorem. We then study generalizations to other loss functions, including the negative log-likelihood for exponential families combined with a strictly convex regularization penalty. The results in this general setting are based on more “indirect” arguments as well as on concentration inequalities for maxima of empirical processes. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000204815Publication status
publishedExternal links
Journal / series
Sankhya AVolume
Pages / Article No.
Publisher
Springer IndiaSubject
Concentration; Density estimation; Empirical process; Empirical risk minimization; Normal sequence model; Penalized least squaresOrganisational unit
03717 - van de Geer, Sara (emeritus) / van de Geer, Sara (emeritus)
Notes
It was possible to publish this article open access thanks to a Swiss National Licence with the publisher.More
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