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 the optimization problem using 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, we define a differentiable approximation of an orientation matrix parameterized by a single priority vector. Differently from previous work, our parameterization fits a smooth orientation matrix and the resulting acyclic adjacency matrix without evaluating acyclicity at any step. Despite the absence of explicit constraints, 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.

翻译：结构学习问题旨在对由有向无环图（DAG）生成的数据进行拟合，以正确重构其弧（arcs）。在该背景下，可微方法利用有向无环图性质的连续松弛形式对优化问题进行约束或正则化。评估图有向无环性的计算复杂度与节点数呈三次方关系，严重制约了可扩展性。本文提出COSMO方法，一种用于有向无环结构学习的无约束连续优化方案。该方法核心在于定义一种由单一优先级向量参数化的方向矩阵的可微近似。与先前工作不同，本文的参数化方法在无需任何步骤评估有向无环性的前提下，同时拟合光滑方向矩阵及由此产生的有向无环邻接矩阵。尽管不存在显式约束，我们证明COSMO始终收敛于有向无环解。除渐近速度更快外，实证分析表明COSMO在图重构性能上优于现有竞争性结构学习方法。