On the Convergence of Continuous Constrained Optimization for Structure Learning

TL;DR

This paper investigates the convergence properties of continuous constrained optimization methods, specifically ALM and QPM, for structure learning of DAGs, revealing limitations of ALM and establishing convergence guarantees for QPM.

Contribution

It provides the first analysis of ALM's convergence in DAG structure learning, shows QPM's convergence to DAGs under mild conditions, and connects these findings with existing methods.

Findings

ALM does not guarantee convergence to DAGs in practice.

QPM converges to DAG solutions under mild conditions.

Empirical results support theoretical convergence guarantees.

Abstract

Recently, structure learning of directed acyclic graphs (DAGs) has been formulated as a continuous optimization problem by leveraging an algebraic characterization of acyclicity. The constrained problem is solved using the augmented Lagrangian method (ALM) which is often preferred to the quadratic penalty method (QPM) by virtue of its standard convergence result that does not require the penalty coefficient to go to infinity, hence avoiding ill-conditioning. However, the convergence properties of these methods for structure learning, including whether they are guaranteed to return a DAG solution, remain unclear, which might limit their practical applications. In this work, we examine the convergence of ALM and QPM for structure learning in the linear, nonlinear, and confounded cases. We show that the standard convergence result of ALM does not hold in these settings, and demonstrate empirically that its behavior is akin to that of the QPM which is prone to ill-conditioning. We further establish the convergence guarantee of QPM to a DAG solution, under mild conditions. Lastly, we connect our theoretical results with existing approaches to help resolve the convergence issue, and verify our findings in light of an empirical comparison of them.