DAG-GNN: DAG Structure Learning with Graph Neural Networks (1904.10098v1)

Abstract: Learning a faithful directed acyclic graph (DAG) from samples of a joint distribution is a challenging combinatorial problem, owing to the intractable search space superexponential in the number of graph nodes. A recent breakthrough formulates the problem as a continuous optimization with a structural constraint that ensures acyclicity (Zheng et al., 2018). The authors apply the approach to the linear structural equation model (SEM) and the least-squares loss function that are statistically well justified but nevertheless limited. Motivated by the widespread success of deep learning that is capable of capturing complex nonlinear mappings, in this work we propose a deep generative model and apply a variant of the structural constraint to learn the DAG. At the heart of the generative model is a variational autoencoder parameterized by a novel graph neural network architecture, which we coin DAG-GNN. In addition to the richer capacity, an advantage of the proposed model is that it naturally handles discrete variables as well as vector-valued ones. We demonstrate that on synthetic data sets, the proposed method learns more accurate graphs for nonlinearly generated samples; and on benchmark data sets with discrete variables, the learned graphs are reasonably close to the global optima. The code is available at \url{https://github.com/fishmoon1234/DAG-GNN}.

• The paper introduces a deep generative model that uses a variational autoencoder with a novel GNN to effectively learn DAG structures from complex, diverse data.
• It proposes an innovative acyclicity constraint that overcomes limitations of traditional matrix exponential methods on modern deep learning platforms.
• Experimental results show improved performance with lower SHD and FDR scores on both synthetic and benchmark datasets, outperforming methods like DAG-NOTEARS.

Overview of DAG-GNN: DAG Structure Learning with Graph Neural Networks

The paper "DAG-GNN: DAG Structure Learning with Graph Neural Networks" presents a novel approach to learning the structure of directed acyclic graphs (DAGs) from data using graph neural networks (GNNs) within a deep generative modeling framework. This task is notoriously difficult due to the combinatorial complexity involved, where the search space size increases superexponentially with the number of nodes.

Theoretical Contributions and Methodology

The authors build upon the framework introduced by Zheng et al. (2018), which reformulates the DAG structure learning problem as a continuous optimization task with an acyclicity constraint. This transformation allows the use of continuous optimization techniques to navigate the otherwise intractable search space. The work presents key innovations in several areas:

1. Deep Generative Model: The DAG-GNN model employs a variational autoencoder (VAE) to learn complex data distributions. The VAE is parameterized by a novel GNN architecture that captures dependencies among variables in a DAG.
2. Handling of Discrete and Vector-Valued Variables: Unlike traditional methods that are limited to certain types of data distributions, the DAG-GNN is versatile in handling scalar, vector-valued, continuous, and discrete variables. This flexibility is achieved by designing the likelihood functions in the VAE to match the nature of the data.
3. Modified Acyclicity Constraint: The paper proposes an alternative acyclicity constraint suitable for current deep learning platforms, overcoming potential issues with implementing the original matrix exponential approach due to platform limitations.

Numerical Results

The proposed method DAG-GNN is evaluated on both synthetic and real-world datasets. Key findings include:

• Synthetic Data: On synthetic datasets generated by both linear and nonlinear models, DAG-GNN consistently learns more accurate graph structures compared to the state-of-the-art linear model-based method, DAG-NOTEARS. This is evident in improved structural Hamming distance (SHD) and false discovery rate (FDR) scores.
• Benchmark and Application Data: For benchmark discrete datasets such as Child, Alarm, and Pigs, the DAG-GNN performs competitively. In practical applications, notably the protein signaling network and causal relation discovery in knowledge bases, DAG-GNN demonstrates its capability in recovering underlying graph structures that align well with known ground truths.

Implications and Future Directions

The DAG-GNN model extends the capacity of graph-based structure learning into deeper and more complex models, which are crucial for capturing nonlinear dependencies and handling diverse data types in real-world applications. This advancement not only improves accuracy but also allows for broader applicability. The proposed approach opens new avenues for exploiting deep learning frameworks in probabilistic graphical models, especially in fields where traditional assumptions do not hold.

Looking forward, future research could focus on enhancing the model's scalability to accommodate larger graphs and improving the computational efficiency of the optimization process. Additionally, further exploration into other types of generative models and network architectures could provide even more powerful methods for structure learning in graphical models.