Mendelson, Shahar2024-03-030039-3223http://hdl.handle.net/1885/315649The 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 Pr(vertical bar f vertical bar >= kappa parallel to f parallel to L-2) >= 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".application/pdfen-AU© Instytut Matematyczny PAN, 2021202110.4064/sm190420-21-112022-10-16