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2015

On concentration for (regularized) empirical risk minimization

van de Geer, Sara, Wainwright, Martin

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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~\cite{chatterjee2014new} 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.

On concentration for (regularized) empirical risk minimization · Around

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Convex Analysis
R.T. Rockafellar · 1970
Earlier work this paper cites.
The Brunn-Minkowski inequality in Gauss space
C. Borell · 1975
Earlier work this paper cites.
Concentration of measure and isoperimetric inequalities in product spaces
M. Talagrand · 1995
Earlier work this paper cites.
Some applications of concentration inequalities to statistics
P. Massart · 2000
Earlier work this paper cites.
The concentration of measure phenomenon , volume 89
M. Ledoux · 2001
Earlier work this paper cites.

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Then

Optimal upper and lower bounds for the true and empirical excess risks in heteroscedastic least-squares regression
A. Saumard · 2012
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Concentration inequalities: A nonasymptotic theory of independence
S. Boucheron, G. Lugosi, and P. Massart · 2013
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A new perspective on least squares under convex constraint
S. Chatterjee · 2014
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Concentration behavior of the penalized least squares estimator, 2015
Original
A. Muro and S. van de Geer · 2015
Closest in time.

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