  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.
