DAG Learning on the Permutahedron

TL;DR

This paper introduces a continuous optimization method for learning DAGs by optimizing over the Permutahedron, enabling valid, modular, and end-to-end learning of DAG structures from observational data.

Contribution

It presents a novel framework that directly optimizes over the space of DAGs using permutation polytopes, improving validity and flexibility over existing relaxations.

Findings

Outperforms existing methods on protein-signaling data

Achieves better SID and SHD metrics in experiments

Supports flexible edge-optimization procedures

Abstract

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.