揭示了图结构学习中选择时间的精确规律，提出无需真实数据即可预测模型表现的新方法。
Support Selection Beyond Smooth DAG Exactness: Completion Geometry,Score Margins, and Selective Certificates
- 通过几何分析推导出孤立环的精确选择时间，建立理论框架。
- 发现分数边界层对选择时间的影响在ν=0时呈对数形式，ν>0时则改变主导动态。
- 提出无真值依赖的分离统计量，可预测320条轨迹的选择时间，适合因果推断研究者。
光滑无环约束用于判断加权支持是否为有向无环图（DAG），而结构学习关注应如何修改支持。现有分析仅针对特定约束公式揭示退化现象，未能分离光滑精确性本身带来的影响。在DAG边界处，我们证明最小循环补全生成一个包含每个受限泰勒射影的平方自由单项式理想。若最小补全有 $q$ 条边，则向量残差的首项响应阶为 $q$，非负标量为 $2q$。指数级多的常尺度环流形在远离边界时表现出相同的排序缺失现象，适用于 NOTEARS 与 DAGMA。我们推导出孤立环的确切选择时间：当 $Ψ'(h)acksim h^ν$ 时，仅可行性时间 $T_0(\varepsilon)=Θ(\varepsilon^{-(2ν+1)})$；分数间隙在 $ν>0$ 时使主导动力学在 $T_0^{-1}$ 尺度上改变，而 $ν=0$ 时需对数边界层满足 $γT_0\log(1/\varepsilon)\to0$。实验验证该规律，且无真值分离统计量在320条官方 NOTEARS/DAGMA 轨迹上成功预测选择时间（斯皮尔曼相关分别为 -0.52 和 -0.66，置换检验 p<10^{-4}）。对于有限样本，父集置信族与强制相反查询可认证所有种群最优解共享的骨架和未屏蔽碰撞器标签。在320次运行中，所有后悔界均覆盖独立的真值分数审计，3,042个骨架标签和2,396个碰撞器标签均无矛盾于真值最优解，尽管分别有4.4%和5.5%与生成图不一致。这些结果区分了DAG可行性、基于分数的支持选择与因果识别。
原文摘要 · Abstract (English)
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.
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