arXiv：cs.LG· Kihun Rhee·· 4 小时前AI 评分28
体积采样岭回归的精确校准与锐利风险几何
Exact Calibration and Sharp Risk Geometry for Volume-Sampled Ridge Regression
AI 导读
研究从固定设计的恰好 s 个不同行出发的岭回归，响应固定、仅子集随机，给出与全数据岭回归期望拟合唯一匹配的惩罚项，其存在条件为 s 超过目标有效维度。针对平衡符号坐标副本，严格扇区不等式在从维度到行数减一的每个预算下给出锐利风险及全部最大化响应；平衡几何还导出同样本无偏的岭–Horvitz–Thompson 混合估计，具有更低锐利风险与精确均值份额改进边界。
正文
Abstract:We study ridge regression from exactly $s$ distinct rows of a fixed design. Responses are fixed, and only the subset is random. The determinant law and selected ridge fit share one positive definite penalty. Established mean identities and exponential-family duality give the unique penalty that matches a prescribed full-data ridge fit in expectation. It exists exactly when $s$ exceeds the target's effective dimension. Our main result concerns centered covariance risk normalized by full-data penalized loss. For balanced signed coordinate replicas, a strict sector inequality gives the sharp risk and all maximizing responses at every budget from the dimension to one below the row count. This holds for any nonzero positive semidefinite query. With the target and query fixed, the maximizing response space is unchanged across these budgets. For general designs, we characterize attainment of a leave-one-out envelope. For existing real equiangular tight frames, flat row query energy characterizes when every nonzero residual response maximizes at two deletions. At three deletions, we give the sharp risk and complete maximizing space for isotropic queries, using unequal triangle weights. The balanced geometry yields a same-sample unbiased ridge--Horvitz--Thompson mixture with lower sharp risk and an exact mean-share improvement boundary. Under full recalibration after feature changes, we prove quadratic regret from searching the complete old maximizing space and a query-uniform bound on the mixture's risk gain. The strongest sector inequalities have exact computer-assisted proofs.
| Comments: | 66 pages, 0 figures |
| Subjects: | Statistics Theory (math.ST); Machine Learning (cs.LG); Machine Learning (stat.ML) |
| Cite as: | arXiv:2610.07721 [math.ST] |
| (or arXiv:2610.07721v1 [math.ST] for this version) | |
| https://doi.org/10.48550/arXiv.2610.07721 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Kihun Rhee [view email]
[v1]
Tue, 6 Oct 2026 04:16:38 UTC (78 KB)
来源：arXiv：cs.LG · arxiv.org