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

TL;DR

This paper establishes the identifiability of the true DAG as the minimum-trace DAG under weakly increasing error variances and demonstrates that a hill climbing algorithm with R2R neighborhood avoids strict local optima, supported by simulations.

Contribution

It extends the identifiable conditions for the minimum-trace DAG and proves the absence of strict local optima in hill climbing with R2R neighborhood.

Findings

True DAG is identifiable as the minimum-trace DAG under weakly increasing error variances.

Hill climbing with R2R neighborhood has no strict local optima.

Simulations confirm the theoretical results with few weak local optima.

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.