超越平滑DAG精确性的支持选择：完备几何、得分边际与选择性证书
Support Selection Beyond Smooth DAG Exactness: Completion Geometry,Score Margins, and Selective Certificates
- School of Computer Science and Engineering, University of Science and Technology of China（中国科学技术大学计算机科学与工程学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对超越平滑DAG精确性的支持选择，推导了孤立环精确选择时间，验证了相关规律，提出无真值分离统计量预测选择时间，用父集置信族等证明标签，区分DAG可行性等内容。
AI中文摘要：
平滑无环约束可判断加权支持是否为有向无环图（DAG），而结构学习则需确定应进行何种支持变更。现有分析针对特定约束公式确立了退化性，但未明确平滑精确性本身会带来何种结果。在DAG边界处，我们证明最小环完备生成无平方单项式理想，该理想包含精确表示的所有受限泰勒喷流。若最小完备包含q条边，则向量残差的首次可能响应阶为q，非负标量的首次可能响应阶为2q。指数级多的常数尺度循环流形在NOTEARS和DAGMA中表现出相同的无法从边界区分的特性。我们推导了孤立环的精确选择时间：当Ψ'(h)∝h^ν时，仅可行性的时间为T₀(ε)=Θ(ε^-(2ν+1))；对于ν>0，得分边际在T₀⁻¹尺度改变主导动态，而ν=0时存在对数边界层，要求γT₀log(1/ε)→0。实验验证了该规律，且无真值的分离统计量可预测320条官方NOTEARS/DAGMA轨迹的选择时间（斯皮尔曼相关系数分别为-0.52和-0.66，置换检验p值<10⁻⁴）。对于有限样本，父集置信族与强制对立查询可证明冻结得分的所有总体最优解所共有的骨架和无遮蔽碰撞体标签。在320次运行中，所有后悔界均覆盖独立的 oracle 得分审计，3042个已证明骨架标签和2396个碰撞体标签均未与oracle得分最优解冲突，尽管分别有4.4%和5.5%与生成图不一致。这些结果将DAG可行性、基于得分的支持选择与因果识别区分开来。
英文摘要：
Smooth acyclicity constraints answer whether a weighted support is a DAG, whereas structure learning asks which support change should be made. Existing analyses establish degeneracy for particular constraint formulas but do not isolate what follows from smooth exactness itself. At a DAG boundary, we show that minimal cycle completions generate a squarefree monomial ideal containing every restricted Taylor jet of an exact representation. If the smallest completion has $q$ edges, the first possible response has order $q$ for a vector residual and $2q$ for a nonnegative scalar. Exponentially many constant-scale cyclic manifolds exhibit the same lack of ranking away from the boundary for NOTEARS and DAGMA. We derive the exact selection time for an isolated cycle. When $Ψ'(h)\asymp h^ν$, the feasibility-only time is $T_0(\varepsilon)=Θ(\varepsilon^{-(2ν+1)})$; a score margin changes the leading dynamics at scale $T_0^{-1}$ for $ν>0$, while $ν=0$ has a logarithmic boundary layer requiring $γT_0\log(1/\varepsilon)\to0$. Experiments verify this law, and a truth-free separation statistic predicts selection time on 320 official NOTEARS/DAGMA trajectories (Spearman $-0.52$ and $-0.66$, permutation $p<10^{-4}$). For finite samples, a parent-set confidence family and forced-opposite queries certify skeleton and unshielded-collider labels shared by every population optimum of a frozen score. Across 320 runs, every regret bound covers an independent oracle-score audit. None of 3,042 certified skeleton or 2,396 collider labels disagrees with the oracle-score optimum, although 4.4% and 5.5%, respectively, disagree with the generating graph. These results separate DAG feasibility, score-based support selection, and causal identification.
