Scale-Sensitive Shattering: Learnability and Evaluability at Optimal Scale
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 an...
| Authors | Shashaank Aiyer, Yishay Mansour, Shay Moran et al. |
|---|---|
| Primary category | cs.LG |
| Published | 13 May 2026 |
| Last updated at source | 29 September 2026 |
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