arXiv：cs.LG（机器学习，全量分类）· Zhongxuan Liu, Hongzhi Wang·· 14 小时前AI 评分36
加权数据选择：尖锐上半与五维定律
Weighted Data Selection: Sharp Upper-Half and Five-Dimensional Laws
AI 导读
针对最小范数学习器的有限加权最小二乘，研究证明在 ⌈3d/2⌉≤n≤2d-1 范围内风险精确满足 Γ_d(n)=3-n/d，且该保证覆盖所有观测特征秩。在 (d,n)=(5,6) 的小预算下，进一步证明 Γ_5(6)=11/5，与单纯形块预测一致。完整的数据集级上界与尖锐性构造已在 Lean 4 中验证。
正文
Abstract:How much risk does a small reweighted training support retain? For finite weighted least squares with the minimum-norm learner, we prove the exact law $\Gamma_d(n)=3-n/d$ throughout $\lceil3d/2\rceil\leq n\leq2d-1$. The guarantee covers every observed feature rank and uses selections that preserve the full feature span. Balanced simplex anchors reduce dimension; positive-weight lifting and independent-line compression close the risk bound. Shifted coordinate pairs attain the matching lower bound. The complete dataset-level upper bound and sharpness construction are verified in Lean 4. At the smaller budget $(d,n)=(5,6)$, we also prove $\Gamma_5(6)=11/5$, matching the simplex-block prediction from $5=3+2$ over arbitrary interacting configurations. Circuit covers, comparison second moments, and circuit-plane probabilities give the sharp excess $6/5$, while polar-face geometry resolves shared rank-three circuits. The general simplex-block frontier connects these laws within the intermediate-budget selection problem.
| Comments: | 35 pages, 2 figures; supplementary verification code included |
| Subjects: | Machine Learning (stat.ML); Machine Learning (cs.LG) |
| Cite as: | arXiv:2610.00101 [stat.ML] |
| (or arXiv:2610.00101v1 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2610.00101 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Hongzhi Wang [view email]
[v1]
Tue, 8 Sep 2026 20:44:50 UTC (1,630 KB)
来源：arXiv：cs.LG（机器学习，全量分类） · arxiv.org