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Inequalities for the Centred Empirical Risk and Its Derivative

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Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 2159)

Abstract

In Theorems 7.1 and 7.2 oracle inequalities for norm-penalized empirical risk minimizers (or M-estimators) were shown. These inequalities require a probability bound for the dual norm of the empirical process. Here such bounds are provided for. The models and methods studied are exponential families, projection estimators and generalized linear models.

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Notes

  1. 1.

    Note that for A = (A 1, , A p ), it holds true that \(\varLambda _{\varOmega _{{\ast}}}(A) \leq \vert \vert \vert A\vert \vert \vert _{\varOmega _{{\ast}}} =:\max _{k}\varOmega _{{\ast}}(A_{k})\).

References

  • S. van de Geer, Empirical Processes in M-Estimation (Cambridge University Press, Cambridge, 2000)

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© 2016 Springer International Publishing Switzerland

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van de Geer, S. (2016). Inequalities for the Centred Empirical Risk and Its Derivative. In: Estimation and Testing Under Sparsity. Lecture Notes in Mathematics(), vol 2159. Springer, Cham. https://doi.org/10.1007/978-3-319-32774-7_10

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