Scale-Sensitive Shattering: Learnability and Evaluability at Optimal Scale

ArXi:2605.13684v1 Announce Type: new 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 $\gamma>0$, uniform convergence at scale $\gamma$, agnostic learnability at scale $\gamma/2$, and finiteness of the fat-shattering dimension at every scale $\gamma'>\gamma$ are equivalent. This resolves a question by Anthony and Bartlett (Cambridge Uni.