Constraint-Free Structure Learning with Smooth Acyclic Orientations

The structure learning problem consists of fitting data generated by a Directed Acyclic Graph (DAG) to correctly reconstruct its arcs. In this context, differentiable approaches constrain or regularize an optimization problem with a continuous relaxation of the acyclicity property. The computational cost of evaluating graph acyclicity is cubic on the number of nodes and significantly affects scalability. In this paper, we introduce COSMO, a constraint-free continuous optimization scheme for acyclic structure learning. At the core of our method lies a novel differentiable approximation of an orientation matrix parameterized by a single priority vector. Differently from previous works, our parameterization fits a smooth orientation matrix and the resulting acyclic adjacency matrix without evaluating acyclicity at any step. Despite this absence, we prove that COSMO always converges to an acyclic solution. In addition to being asymptotically faster, our empirical analysis highlights how COSMO performance on graph reconstruction compares favorably with competing structure learning methods.