We propose a continuous optimization framework for discovering a latent directed acyclic graph (DAG) from observational data. Our approach optimizes over the polytope of permutation vectors, the so-called Permutahedron, to learn a topological ordering. Edges can be optimized jointly, or learned conditional on the ordering via a non-differentiable subroutine. Compared to existing continuous optimization approaches our formulation has a number of advantages including: 1. validity: optimizes over exact DAGs as opposed to other relaxations optimizing approximate DAGs; 2. modularity: accommodates any edge-optimization procedure, edge structural parameterization, and optimization loss; 3. end-to-end: either alternately iterates between node-ordering and edge-optimization, or optimizes them jointly. We demonstrate, on real-world data problems in protein-signaling and transcriptional network discovery, that our approach lies on the Pareto frontier of two key metrics, the SID and SHD.

翻译：我们提出了一种连续优化框架，用于从观测数据中发现潜在的有向无环图（DAG）。该方法在置换向量的多面体（即所谓的置换体）上进行优化，以学习拓扑排序。边结构可联合优化，或通过不可微子程序基于排序条件进行学习。与现有连续优化方法相比，本公式具有以下优势：1. 有效性：直接优化精确DAG，而非其他松弛方法中优化的近似DAG；2. 模块化：可兼容任何边优化过程、边结构参数化方案及优化损失函数；3. 端到端：既可交替迭代节点排序与边优化，亦可进行联合优化。我们在蛋白质信号传导和转录网络发现的真实数据问题中证明，该方法在结构干预距离（SID）与结构汉明距离（SHD）两项关键指标上处于帕累托前沿。