Scale-Sensitive Shattering: Learnability and Evaluability at Optimal Scale

2026-05-13

AI Summary

This work investigates the uniform convergence and learnability of real-valued function classes at an arbitrary scale γ, establishing a scale-sensitive fundamental theorem of PAC learning: uniform convergence, agnostic learnability at scale γ/2, and finite fat-shattering dimension at all larger scales are equivalent. By directly controlling empirical ℓ∞ covering numbers and bypassing traditional packing-number arguments, the study refutes Long’s conjecture that a factor-of-two gap in scale is unavoidable, yielding for the first time a tight characterization without multiplicative loss. It resolves open problems on metric entropy posed by Alon et al. and Rudelson–Vershynin, provides a sharp O(log²n) metric entropy bound at scale γ/2 (sometimes unimprovable), an O(log n) bound at scale 2γ, and establishes a sharp dichotomy for the 3-weak learnability of integral probability metrics (IPMs).

Abstract

We study the optimal scale at which real-valued function classes exhibit uniform convergence and learnability. Our main result establishes a scale-sensitive generalization of the fundamental theorem of PAC learning: for every bounded real-valued class and every $γ>0$, uniform convergence at scale $γ$, agnostic learnability at scale $γ/2$, and finiteness of the fat-shattering dimension at every scale $γ'>γ$ are equivalent. This resolves a question by Anthony and Bartlett (Cambridge Univ. Press 1999) on the precise scales governing learnability, refuting a conjecture attributed there to Phil Long that a multiplicative 2-factor gap is unavoidable, and improves the upper bounds of Bartlett and Long (JCSS 1998), which incur such a loss. The key technical ingredient is a direct bound on empirical $\ell_\infty$ covering numbers, avoiding the standard detour through packing numbers. As a consequence, we obtain sharp asymptotic metric-entropy bounds in terms of the fat-shattering scale $γ$: an $O(\log^2 n)$ bound holds already at scale $γ/2$, while an $O(\log n)$ bound holds at scale $2γ$. We further show that the $O(\log^2 n)$ bound is sometimes tight. These results resolve open questions by Alon et al. (JACM 1997) and Rudelson and Vershynin (Ann. of Math. 2006). As an application, we establish a sharp dichotomy for bounded integral probability metrics: every such IPM is either estimable or cannot be weakly evaluated within any multiplicative factor $c<3$, while $3$-weak evaluability always holds, resolving an open question from Aiyer et al. (ICML 2026). We also highlight several open questions on quantitative sample complexity and evaluability.

Problem

• scale-sensitive
• learnability
• uniform convergence
• fat-shattering dimension
• PAC learning

Innovation

• scale-sensitive learning
• fat-shattering dimension
• uniform convergence
• covering numbers
• integral probability metrics