On the Convergence of Continuous Constrained Optimization for Structure Learning
This work clarifies the theoretical convergence guarantees for optimization methods used in structure learning of DAGs, which is important for researchers and practitioners relying on these methods to ensure valid DAG solutions.
This paper investigates the convergence properties of Augmented Lagrangian Method (ALM) and Quadratic Penalty Method (QPM) for continuous constrained optimization in structure learning of Directed Acyclic Graphs (DAGs). It finds that the standard convergence result for ALM does not hold in this context, and empirically shows its behavior is similar to QPM, which is prone to ill-conditioning. The paper establishes a convergence guarantee for QPM to a DAG solution under mild conditions.
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.