# Multi-task Learning of Order-Consistent Causal Graphs

arXiv: https://arxiv.org/abs/2111.02545 · HTML: https://mlsanity.com/paper/2111.02545/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 57 references that are in the map.
- First posted on arXiv 3 Nov 2021.

## Builds on (most cited first, 13 of 13)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Gradient-Based Neural DAG Learning](https://mlsanity.com/paper/1906.02226/) (2019)
- [Learning Sparse Nonparametric DAGs](https://mlsanity.com/paper/1909.13189/) (2019)
- [DYNOTEARS: Structure Learning from Time-Series Data](https://mlsanity.com/paper/2002.00498/) (2020)
- [Causal Discovery with Reinforcement Learning](https://mlsanity.com/paper/1906.04477/) (2019)
- [Masked Gradient-Based Causal Structure Learning](https://mlsanity.com/paper/1910.08527/) (2019)
- [Node-Based Learning of Multiple Gaussian Graphical Models](https://mlsanity.com/paper/1303.5145/) (2013)
- [Integer Programming for Learning Directed Acyclic Graphs from Continuous Data](https://mlsanity.com/paper/1904.10574/) (2019)
- [Learning Directed Acyclic Graphs with Penalized Neighbourhood Regression](https://mlsanity.com/paper/1511.08963/) (2015)
- [High-Dimensional Joint Estimation of Multiple Directed Gaussian Graphical Models](https://mlsanity.com/paper/1804.00778/) (2018)
- [Direct Learning with Guarantees of the Difference DAG Between Structural Equation Models](https://mlsanity.com/paper/1906.12024/) (2019)
- [Bayesian joint inference for multiple directed acyclic graphs](https://mlsanity.com/paper/2008.06190/) (2020)

## Built on by (most cited first, 10 of 10)

- [Structure Learning with Continuous Optimization: A Sober Look and Beyond](https://mlsanity.com/paper/2304.02146/) (2023)
- [MM-DAG: Multi-task DAG Learning for Multi-modal Data -- with Application for Traffic Congestion Analysis](https://mlsanity.com/paper/2306.02831/) (2023)
- [Finding Counterfactually Optimal Action Sequences in Continuous State Spaces](https://mlsanity.com/paper/2306.03929/) (2023)
- [psiDAG: Projected Stochastic Approximation Iteration for DAG Structure Learning](https://mlsanity.com/paper/2410.23862/) (2024)
- [Learning Causal Graphs at Scale: A Foundation Model Approach](https://mlsanity.com/paper/2506.18285/) (2025)
- [Revisiting Differentiable Structure Learning: Inconsistency of ℓ1 Penalty and Beyond](https://mlsanity.com/paper/2410.18396/) (2024)
- [Consistency evaluation of benchmarks used for causal discovery](https://mlsanity.com/paper/2606.01789/) (2026)
- [Differentiable and Transportable Structure Learning](https://mlsanity.com/paper/2206.06354/) (2022)
- [Structural transfer learning of non-Gaussian DAG](https://mlsanity.com/paper/2310.10239/) (2023)
- [Invariant Causal Prediction with Local Models](https://mlsanity.com/paper/2401.05218/) (2024)

## Abstract (arXiv)

We consider the problem of discovering K related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a l1/l2-regularized maximum likelihood estimator (MLE) for learning K linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.