Weighted Data Selection: Sharp Upper-Half and Five-Dimensional Laws
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 Γd(n)=3−n/d throughout ⌈3d/2⌉≤ n≤2d−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 Γ5(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.
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BibTeX
@misc{liu2026weighted,
title = {{Weighted Data Selection: Sharp Upper-Half and Five-Dimensional Laws}},
author = {Zhongxuan Liu and Hongzhi Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00101},
eprint = {2610.00101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00101},
}
RIS
TY - UNPB
TI - Weighted Data Selection: Sharp Upper-Half and Five-Dimensional Laws
AU - Zhongxuan Liu
AU - Hongzhi Wang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00101
UR - https://arxiv.org/abs/2610.00101
AB - 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 Γ_d(n)=3−n/d throughout ⌈3d/2⌉≤ n≤2d−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 Γ_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.
ER -