Extending the small-ball method

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 is, $Pr(|f| \geq \kappa\|f\|_{L_2}) \geq \delta$ 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'. As applications, we study the performance of empirical risk minimization in learning problems involving bounded subsets of $L_p$ that satisfy a Bernstein condition, and of the tournament procedure in problems involving bounded subsets of $L_\infty$.