发表机构
Harbin Institute of Technology（哈尔滨工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对最小范数加权最小二乘，证明重加权训练支持集的风险精确定律，并给出锐利上界与下界，在中间预算下用单纯形块统一连接不同维度结果。
AI 中文摘要
一个经过小幅重加权后的训练支持集能保留多少风险？对于使用最小范数学习器的有限加权最小二乘问题，我们在 $\nceil{3d/2}\nleq n\nleq 2d-1$ 范围内证明了精确定律 $\nGamma_d(n)=3-n/d$。该保证覆盖了所有观测到的特征秩，并使用保留完整特征跨度的选择。平衡单纯形锚点降低了维度；正权重提升和独立线压缩收紧了风险界限。平移坐标对达到了匹配的下界。完整的数据集级上界和锐利性构造已在 Lean 4 中验证。在较小的预算 $(d,n)=(5,6)$ 下，我们还证明了 $\nGamma_5(6)=11/5$，与来自 $5=3+2$ 的单纯形块预测在任意交互配置上相匹配。电路覆盖、比较二阶矩和电路平面概率给出了锐利的超额 $6/5$，而极面几何解决了共享秩三电路的问题。一般的单纯形块前沿在中间预算选择问题中连接了这些定律。
英文摘要
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 $5=3+2$ 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.
Comments35 pages, 2 figures; supplementary verification code included
