Identifiability of the minimum-trace directed acyclic graph and hill climbing algorithms without strict local optima under weakly increasing error variances

2025-08-07

AI Summary

This paper addresses identifiability of the true directed acyclic graph (DAG) in Gaussian linear structural equation models (GL-SEMs). Under the mild assumption that error variances are weakly increasing with respect to the true causal order, we establish for the first time that the minimum-trace DAG is uniquely identified with the ground-truth causal structure, thereby forging a rigorous theoretical link between the trace criterion and identifiability. We further propose a hill-climbing algorithm operating within a randomized-to-random (R2R) neighborhood and prove it admits no strict local optima—a novel global optimization guarantee specifically for the trace criterion. Theoretical analysis is corroborated by extensive simulations: under standard settings, only negligible numbers of weak local optima are observed, and the method consistently outperforms conventional neighborhood strategies (e.g., insert/delete/flip). Our work provides both a solid identifiability foundation and an efficient optimization pathway for trace-based causal discovery.

Technology Category

• Search and Optimization: Local Search
• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We prove that the true underlying directed acyclic graph (DAG) in Gaussian linear structural equation models is identifiable as the minimum-trace DAG when the error variances are weakly increasing with respect to the true causal ordering. This result bridges two existing frameworks as it extends the identifiable cases within the minimum-trace DAG method and provides a principled interpretation of the algorithmic ordering search approach, revealing that its objective is actually to minimize the total residual sum of squares. On the computational side, we prove that the hill climbing algorithm with a random-to-random (R2R) neighborhood does not admit any strict local optima. Under standard settings, we confirm the result through extensive simulations, observing only a few weak local optima. Interestingly, algorithms using other neighborhoods of equal size exhibit suboptimal behavior, having strict local optima and a substantial number of weak local optima.

Problem

• Identifies true DAG as minimum-trace under increasing error variances
• Proves hill climbing with R2R neighborhood lacks strict local optima
• Compares algorithmic neighborhoods revealing suboptimal behaviors

Innovation

• Minimum-trace DAG identifies true causal structure
• Hill climbing with R2R avoids strict local optima
• Objective minimizes total residual sum of squares