PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Extending the scope of the small-ball method

Volume 256 / 2021

Shahar Mendelson Studia Mathematica 256 (2021), 147-167 MSC: Primary 60G99. DOI: 10.4064/sm190420-21-11 Published online: 19 June 2020

Abstract

The small-ball method was introduced as a way of obtaining a high probability, isomorphic lower bound on the quadratic empirical process, under weak assumptions on the indexing class. The key assumption was that class members satisfy a uniform small-ball estimate: that ${\rm Pr}(|f| \geq \kappa \|f\|_{L_2}) \geq \delta $ for given constants $\kappa $ and $\delta $.

Here we extend the small-ball method and obtain a high probability, almost-isometric (rather than isomorphic) lower bound on the quadratic empirical process. The scope of the result is considerably wider than the small-ball method: there is no need for class members to satisfy a uniform small-ball condition, and moreover, motivated by the notion of tournament learning procedures, the result is stable under a “majority vote”.

Authors

  • Shahar MendelsonMathematical Science Institute The Australian National University Canberra, Australia
    and
    LPSM, Sorbonne University
    Paris, France
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image