体积采样岭回归的精确校准与尖锐风险几何
Exact Calibration and Sharp Risk Geometry for Volume-Sampled Ridge Regression
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中文总结 AI 辅助
本文研究体积采样岭回归的精确校准与风险几何，提出唯一惩罚匹配全数据岭拟合，并刻画各预算下的尖锐风险与最大化响应空间，具有理论证明与实用意义。
中文摘要 AI 辅助
我们研究从固定设计的恰好 $s$ 个不同行进行岭回归。响应是固定的，只有子集是随机的。行列式法则和选定的岭拟合共享一个正定惩罚。已建立的均值恒等式和指数族对偶性给出了与期望中规定的全数据岭拟合相匹配的唯一惩罚。它恰好存在于 $s$ 超过目标的有效维度时。我们的主要结果涉及以全数据惩罚损失归一化的中心协方差风险。对于平衡符号坐标副本，严格的扇形不等式给出了从维度到行数减一的每个预算下的尖锐风险和所有最大化响应。这适用于任何非零正半定查询。当目标和查询固定时，最大化响应空间在这些预算中保持不变。对于一般设计，我们刻画了留一包络的达到。对于现有的实等角紧框架，平坦行查询能量刻画了何时每个非零残差响应在两次删除时最大化。在三次删除时，我们给出了各向同性查询的尖锐风险和完整最大化空间，使用不等三角形权重。平衡几何产生了一个同样本无偏的岭-霍维茨-汤普森混合，具有较低的尖锐风险和精确的均值份额改进边界。在特征变化后的完全重新校准下，我们证明了搜索完整旧最大化空间的二次遗憾和对混合风险增益的查询均匀界限。最强的扇形不等式有精确的计算机辅助证明。
英文摘要
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.
