From graphs to DAGs: a low-complexity model and a scalable algorithm

TL;DR

This paper introduces LoRAM, a low-complexity, scalable algorithm for learning DAGs that reduces computational complexity from cubic to quadratic, enabling efficient handling of large graphs with minimal accuracy loss.

Contribution

LoRAM combines low-rank matrix factorization with sparsification to efficiently optimize DAGs, significantly reducing computational complexity compared to existing methods.

Findings

LoRAM achieves quadratic complexity, scalable to thousands of nodes.

It offers orders of magnitude efficiency gains over state-of-the-art methods.

Moderate accuracy loss is observed with sparse matrices, with low sensitivity to rank choice.

Abstract

Learning directed acyclic graphs (DAGs) is long known a critical challenge at the core of probabilistic and causal modeling. The NoTears approach of (Zheng et al., 2018), through a differentiable function involving the matrix exponential trace $\mathrm{tr}(\exp(\cdot))$, opens up a way to learning DAGs via continuous optimization, though with a $O(d^3)$ complexity in the number $d$ of nodes. This paper presents a low-complexity model, called LoRAM for Low-Rank Additive Model, which combines low-rank matrix factorization with a sparsification mechanism for the continuous optimization of DAGs. The main contribution of the approach lies in an efficient gradient approximation method leveraging the low-rank property of the model, and its straightforward application to the computation of projections from graph matrices onto the DAG matrix space. The proposed method achieves a reduction from a cubic complexity to quadratic complexity while handling the same DAG characteristic function as NoTears, and scales easily up to thousands of nodes for the projection problem. The experiments show that the LoRAM achieves efficiency gains of orders of magnitude compared to the state-of-the-art at the expense of a very moderate accuracy loss in the considered range of sparse matrices, and with a low sensitivity to the rank choice of the model's low-rank component.