Support Selection Beyond Smooth DAG Exactness: Completion Geometry,Score Margins, and Selective Certificates

2026-08-08

AI Summary

This work addresses the challenge in existing smooth acyclicity-constrained structure learning methods, which struggle to decouple support set selection from DAG feasibility and lack theoretical characterization of how smoothing affects exactness. By analyzing minimal cycle-completion structures at the DAG boundary, the paper establishes a novel connection between the generated square-free monomial ideal and the Taylor expansion of the exact acyclicity constraint. This insight yields theoretical links among support selection, geometric structure, score margin, and selective certification. Building on this foundation, the authors propose a ground-truth-free separation statistic to predict selection timing and develop a parent-set confidence family with a forced reverse-query mechanism to certify both the skeleton and unshielded colliders. Experiments across 320 NOTEARS/DAGMA trajectories validate the selection-time law (Spearman correlation −0.52 to −0.66, p<10⁻⁴), with 3,042 certified skeletons and 2,396 collider labels matching the globally optimal solutions.

Abstract

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.

Problem

• DAG structure learning
• smooth acyclicity constraints
• support selection
• causal identification
• score-based optimization

Innovation

• cycle completion
• score margin
• selective certificate
• smooth DAG exactness
• structure learning