发表机构
Politecnico di Torino（都灵理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定场景优化等方法中风险-复杂度定律的边界机制，推导随机边界大小下的精确定律，覆盖多种校准与认证场景，给出条件概率证书及不可行结果。
AI 中文摘要
场景优化、共形预测及相关无分布认证方法利用有限样本构建决策或预测集，为新观测提供违反风险保证。在若干经典场景中，条件违反风险遵循精确的beta定律，其尾部具有beta-二项式表示，参数为支撑、校准或压缩维度。本文确定了这些公式背后的确定性边界机制，并在观测边界大小为随机时推导了相应定律。决策规则由未来观测的接受集及选择负责该集的样本点的边界映射表示；当保留全样本边界时恰好接受留存样本，且可删除非边界样本而不改变该边界时，所得对称为“恰当投影边界方案”。对于每个此类方案，给定观测边界大小的条件违反风险定律由边界的跨样本复杂度轮廓确定：稳定轮廓产生常规beta定律，而变化轮廓产生精确的轮廓修正。该框架涵盖标量顺序统计量校准、支撑重构场景程序、级联支撑移除证书、坐标方向包络及向量得分的帕累托前沿校准，还产生条件概率证书及解释仅观测复杂度不足的不可行结果。
英文摘要
Scenario optimization, conformal prediction, and related distribution-free certification methods use finite samples to construct decisions or prediction sets with violation-risk guarantees for fresh observations. In several classical settings, the conditional violation risk follows an exact beta law, whose tail has a beta-binomial representation and whose parameter is a support, calibration, or compression dimension. This paper identifies the deterministic boundary mechanism behind these formulas and derives the corresponding law when the observed boundary size is random. A decision rule is represented by an acceptance set for future observations, together with a boundary map selecting the sample points responsible for that set. The resulting pair is called a {\em proper projective boundary scheme} when held-out samples are accepted precisely if the full-sample boundary is retained, and accepted non-boundary samples can be deleted without changing that boundary. For every such scheme, the conditional law of the violation risk given the observed boundary size is determined by the boundary's cross-sample complexity profile. A stable profile yields the usual beta law, whereas a varying profile produces an exact profile correction. The framework covers scalar order-statistic calibration, support-reconstructive scenario programs, cascaded support-removal certificates, coordinatewise envelopes, and Pareto-frontier calibration with vector scores. It also yields conditional probabilistic certificates and a no-go result explaining why observed complexity alone is insufficient.
