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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


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”.


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

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