Weighted Data Selection: Sharp Upper-Half and Five-Dimensional Laws
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 $\lceil3d/2\rceil\leq n\leq2d-1$. 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 $Γ_5(6)=11/5$, matching the simplex-block prediction from $...
| Authors | Zhongxuan Liu, Hongzhi Wang |
|---|---|
| Primary category | stat.ML |
| Published | 8 September 2026 |
| Last updated at source | 8 September 2026 |
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