A Fine-Grained Understanding of Uniform Convergence for Halfspaces
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914539235180544 |
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| author | Kontorovich, Aryeh Larsen, Kasper Green |
| author_facet | Kontorovich, Aryeh Larsen, Kasper Green |
| contents | We study the fine-grained uniform convergence behavior of halfspaces beyond worst-case VC bounds. For inhomogeneous halfspaces in $\mathbb{R}^d$ with $d\ge 2$, we show that standard first-order VC bounds are essentially tight: even consistent hypotheses can incur population error $Θ(d\ln(n/d)/n)$, and in the agnostic setting the deviation scales as $\sqrt{τ\ln(1/τ)}$ at true error $τ$. In contrast, homogeneous halfspaces in $\mathbb{R}^2$ exhibit a markedly different behavior. In the realizable case, every hypothesis consistent with the sample has error $O(1/n)$. In the agnostic case, we prove a bandwise, log-free deviation bound on each dyadic risk band via a critical-wedge localization argument. Unioning over bands incurs only a $\ln\ln n$ overhead, and we establish a matching lower bound showing this overhead is unavoidable. Together, these results give a fine-grained and nearly complete picture of uniform convergence for halfspaces, revealing sharp dimensional and structural thresholds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06004 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Fine-Grained Understanding of Uniform Convergence for Halfspaces Kontorovich, Aryeh Larsen, Kasper Green Machine Learning Artificial Intelligence Statistics Theory We study the fine-grained uniform convergence behavior of halfspaces beyond worst-case VC bounds. For inhomogeneous halfspaces in $\mathbb{R}^d$ with $d\ge 2$, we show that standard first-order VC bounds are essentially tight: even consistent hypotheses can incur population error $Θ(d\ln(n/d)/n)$, and in the agnostic setting the deviation scales as $\sqrt{τ\ln(1/τ)}$ at true error $τ$. In contrast, homogeneous halfspaces in $\mathbb{R}^2$ exhibit a markedly different behavior. In the realizable case, every hypothesis consistent with the sample has error $O(1/n)$. In the agnostic case, we prove a bandwise, log-free deviation bound on each dyadic risk band via a critical-wedge localization argument. Unioning over bands incurs only a $\ln\ln n$ overhead, and we establish a matching lower bound showing this overhead is unavoidable. Together, these results give a fine-grained and nearly complete picture of uniform convergence for halfspaces, revealing sharp dimensional and structural thresholds. |
| title | A Fine-Grained Understanding of Uniform Convergence for Halfspaces |
| topic | Machine Learning Artificial Intelligence Statistics Theory |
| url | https://arxiv.org/abs/2605.06004 |