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On Concentration for (Regularized) Empirical Risk Minimization

Author

Listed:
  • Sara Geer

    (Seminar for Statistics, ETH Zürich)

  • Martin J. Wainwright

    (University of California)

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.

Suggested Citation

  • Sara Geer & Martin J. Wainwright, 2017. "On Concentration for (Regularized) Empirical Risk Minimization," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 159-200, August.
    • Handle: RePEc:spr:sankha:v:79:y:2017:i:2:d:10.1007_s13171-017-0111-9
    DOI: 10.1007/s13171-017-0111-9
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    Cited by:

    1. Stéphane Boucheron, 2017. "Discussion of “concentration for (regularized) empirical risk minimization” by Sara van de Geer and Martin Wainwright," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 201-207, August.

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