  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 $Pr(|f| \geq
\kappa\|f\|_{L_2}) \geq \delta$ 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'.

Machine Learning