A Gapped Scale-Sensitive Dimension and Lower Bounds for Offset Rademacher Complexity

Zeyu Jia; Yury Polyanskiy; Alexander Rakhlin

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arXiv · 2509.20618

A Gapped Scale-Sensitive Dimension and Lower Bounds for Offset Rademacher Complexity

Zeyu Jia·Yury Polyanskiy·Alexander Rakhlin

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Abstract

We study gapped scale-sensitive dimensions of a function class in both sequential and non-sequential settings. We demonstrate that covering numbers for any uniformly bounded class are controlled above by these gapped dimensions, generalizing the results of \cite{anthony2000function,alon1997scale}. Moreover, we show that the gapped dimensions lead to lower bounds on offset Rademacher averages, thereby strengthening existing approaches for proving lower bounds on rates of convergence in statistical and online learning.

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Zeyu Jia, Yury Polyanskiy, Alexander Rakhlin. 2025-09-24. A Gapped Scale-Sensitive Dimension and Lower Bounds for Offset Rademacher Complexity. https://arxiv.org/abs/2509.20618

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