DAGs with NO TEARS: Continuous Optimization for Structure Learning (1803.01422v2)

Abstract: Estimating the structure of directed acyclic graphs (DAGs, also known as Bayesian networks) is a challenging problem since the search space of DAGs is combinatorial and scales superexponentially with the number of nodes. Existing approaches rely on various local heuristics for enforcing the acyclicity constraint. In this paper, we introduce a fundamentally different strategy: We formulate the structure learning problem as a purely \emph{continuous} optimization problem over real matrices that avoids this combinatorial constraint entirely. This is achieved by a novel characterization of acyclicity that is not only smooth but also exact. The resulting problem can be efficiently solved by standard numerical algorithms, which also makes implementation effortless. The proposed method outperforms existing ones, without imposing any structural assumptions on the graph such as bounded treewidth or in-degree. Code implementing the proposed algorithm is open-source and publicly available at https://github.com/xunzheng/notears.

• The paper introduces a continuous optimization framework that transforms the NP-hard acyclicity constraint into a smooth equality constraint.
• It employs an augmented Lagrangian method to enforce acyclicity effectively, achieving lower Structural Hamming Distance and False Discovery Rates compared to traditional methods.
• The method's simplicity and open-source implementation make it a practical and scalable tool for DAG structure learning across various research domains.

Continuous Optimization for Structure Learning of Directed Acyclic Graphs (DAGs)

The paper "DAGs with NO TEARS: Continuous Optimization for Structure Learning" by Xun Zheng, Bryon Aragam, Pradeep Ravikumar, and Eric P. Xing introduces a novel approach to learning the structure of Directed Acyclic Graphs (DAGs) through continuous optimization. This method leverages a natural acyclicity constraint reformulation, facilitating the use of standard numerical algorithms for efficient DAG learning without combinatorial complexities.

Problem Overview

Learning DAGs is an essential problem in various domains such as biology, genetics, machine learning, and causal inference. Traditional methods face significant challenges due to the combinatorial nature of the search space, making the enforcement of the acyclicity constraint NP-hard. Existing approaches often resort to local heuristics, which can be complex and computationally intensive.

The Proposed Approach

This paper proposes a fundamentally different strategy by converting the discrete, combinatorial optimization problem into a continuous one using an innovative characterization of acyclicity. The new problem formulation maintains the essential properties needed to enforce acyclicity but allows for smooth optimization over real matrices. This transformation results in a continuous program that can be effectively solved using standard optimization techniques, making the implementation straightforward and accessible.

The key idea lies in using a characterization for acyclicity via matrix functions that can be evaluated and optimized in a continuous manner. Specifically, the approach replaces the acyclicity constraint with a smooth equality constraint utilizing matrix exponential operations.

Core Contributions

1. Smooth Acyclicity Characterization: The paper introduces a novel function $h(W)$ involving the matrix exponential $e^{W \circ W}$, where $W$ is the weighted adjacency matrix. This function is shown to be zero if and only if $W$ is acyclic, ensuring an accurate and smooth enforcement of the DAG constraint.
2. Augmented Lagrangian Method: An enhanced optimization method is employed to handle the equality-constrained program efficiently. The augmented Lagrangian approach is adopted to iteratively optimize the objective function while enforcing the acyclicity constraint.
3. Implementation and Practicality: The method is highly practical, requiring minimal lines of code using standard optimization solvers. The authors provide an open-source implementation, making it accessible for broader use.
4. Empirical Evaluation: Extensive experiments demonstrate the efficacy of the proposed method. The algorithm outperforms existing state-of-the-art methods (e.g., GES, PC, and LiNGAM) in different scenarios, particularly when the number of nodes and edges is large.

Numerical Results

The intuitive effectiveness of the method is showcased through qualitative and quantitative analyses. For instance, in structure learning, the proposed method displays superior performance in scenarios with varying noise models and graph structures, achieving lower Structural Hamming Distance (SHD) and False Discovery Rates (FDR) compared to baseline methods such as Fast Greedy Search (FGS).

Figures and tables illustrate the robustness and consistency of parameter estimates as a function of sample size, with improved performance for larger datasets due to the regularization strategies employed. Despite using thresholding to ensure numerical stability and feasibility, the method retains high accuracy in edge detection.

Future Implications

The approach opens several potential developments in AI and ML:

• Scalability: While already effective for moderately sized graphs, further research could enhance scalability to very large datasets.
• Generalization to Other Models: Extending this continuous optimization approach to other forms of graphical models or utilizing different structural sparsity constraints.
• Non-Smooth Scores: Adapting the method to handle non-smooth or discrete score functions, potentially leveraging techniques like Nesterov's smoothing.

Conclusion

This paper presents a significant advancement in the field of structure learning for DAGs by formulating the problem as a continuous optimization task. The proposed $NOTEARS$ algorithm effectively learns the structure and parameters of DAGs, overcoming the traditionally combinatorial hurdles through an elegant mathematical reformulation. The simplicity, efficiency, and superior performance of the approach make it a valuable tool for researchers and practitioners alike.

The source code and resources for replicating the results are available at https://github.com/xunzheng/notears, promoting transparency and encouraging further contributions to this promising approach.