# Extending the scope of the small-ball method

arXiv: https://arxiv.org/abs/1709.00843 · HTML: https://mlsanity.com/paper/1709.00843/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Sep 2017.

## Built on by (most cited first, 2 of 2)

- [Minimum Norm Interpolation via the Local Theory of Banach Spaces: The Role of 2-Uniform Convexity](https://mlsanity.com/paper/2603.28956/) (2026)
- [Subspace Uncertainty and Sharp Sampling Thresholds on the Boolean Cube](https://mlsanity.com/paper/2610.12358/) (2026)

## Abstract (arXiv)

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| ≥ kappa|f|L2) ≥ δ for given constants kappa and δ. 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'.