DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization (2209.08037v3)

Abstract: The combinatorial problem of learning directed acyclic graphs (DAGs) from data was recently framed as a purely continuous optimization problem by leveraging a differentiable acyclicity characterization of DAGs based on the trace of a matrix exponential function. Existing acyclicity characterizations are based on the idea that powers of an adjacency matrix contain information about walks and cycles. In this work, we propose a new acyclicity characterization based on the log-determinant (log-det) function, which leverages the nilpotency property of DAGs. To deal with the inherent asymmetries of a DAG, we relate the domain of our log-det characterization to the set of $\textit{M-matrices}$, which is a key difference to the classical log-det function defined over the cone of positive definite matrices. Similar to acyclicity functions previously proposed, our characterization is also exact and differentiable. However, when compared to existing characterizations, our log-det function: (1) Is better at detecting large cycles; (2) Has better-behaved gradients; and (3) Its runtime is in practice about an order of magnitude faster. From the optimization side, we drop the typically used augmented Lagrangian scheme and propose DAGMA ($\textit{DAGs via M-matrices for Acyclicity}$), a method that resembles the central path for barrier methods. Each point in the central path of DAGMA is a solution to an unconstrained problem regularized by our log-det function, then we show that at the limit of the central path the solution is guaranteed to be a DAG. Finally, we provide extensive experiments for $\textit{linear}$ and $\textit{nonlinear}$ SEMs and show that our approach can reach large speed-ups and smaller structural Hamming distances against state-of-the-art methods. Code implementing the proposed method is open-source and publicly available at https://github.com/kevinsbello/dagma.

• The paper introduces a novel acyclicity characterization using the log-det function and M-matrices to efficiently learn directed acyclic graphs.
• It demonstrates superior cycle detection, stable gradient performance, and runtime improvements compared to traditional methods.
• The optimization framework, validated on both linear and nonlinear SEMs, offers clear advantages for large-scale data applications.

A Comprehensive Analysis of the DAGMA Framework for Learning DAGs

The paper presents a novel approach to the combinatorial problem of learning directed acyclic graphs (DAGs), introducing DAGMA (Directed Acyclic Graphs via M-matrices for Acyclicity). This approach leverages a new acyclicity characterization through the log-determinant (log-det) function, offering an innovative solution to the continuous optimization landscape for DAG learning. This is a marked departure from earlier methods that relied on powers of adjacency matrices for acyclicity checks.

Core Contributions

1. Novel Acyclicity Characterization: The work introduces a differentiation-based acyclicity characterization using the log-det function. The domain of this characterization is linked to M-matrices—matrices of the form $A = sI - B$ where $B \geq 0$ and $s > \rho(B)$. This novel use of M-matrices is crucial for interpreting structure learning for DAGs.
2. Superior Detection of Cycles: Compared to previous methods, the log-det function is better suited for detecting larger cycles, offers more stable gradients, and is computationally more efficient. This makes DAGMA particularly advantageous for large-scale data.
3. Optimization Framework: The paper opts for a central path methodology over the traditional augmented Lagrangian method, ensuring that solutions converge to a DAG without requiring post-hoc thresholding. The central path approach effectively positions the log-det function as a regularizer, akin to classical norm regularizers.
4. Extensive Empirical Validation: The paper validates DAGMA’s effectiveness through extensive experiments spanning linear and nonlinear structural equation models (SEMs). The results indicate significant improvements in run time and structural Hamming distance (SHD) over state-of-the-art methods like NOTEARS and GOLEM.

Theoretical Insights

One of the theoretical pillars of the paper is the invex property of the log-det function, which ensures that all local minima are global, aligning with the function's characteristic that all DAGs are minima. This provides a mathematical guarantee that aligning the optimization with the function's geometrical properties correctly identifies the DAGs.

Practical and Theoretical Implications

• Computational Efficiency: With runtime improvements often an order of magnitude greater than its predecessors, DAGMA presents a compelling case for adoption in large-scale applications.
• Scalability and Robustness: The log-det characterization's stability in detecting cycles and handling gradients positions it as robust across various applications in fields such as bioinformatics, network analysis, and econometrics.
• Methodological Advancements: The paper's methodology may pave the way for further exploration into continuous optimization approaches for combinatorial problems beyond DAGs.

Future Directions

• Extension to Complex Data Structures: Given its foundational approach, extending this method to handle more complex graphical structures such as cyclic graphs or dynamic networks could be of significant value.
• Addressing Hidden Confounders: Further research could account for hidden variables affecting observed variables, currently an assumption not addressed.

In conclusion, the DAGMA framework offers a significant enhancement to the task of learning DAGs by introducing a computationally efficient, mathematically robust method based on the innovative use of M-matrices and the log-det function. The empirical and theoretical advancements laid out in this paper invite further exploration and application in diverse areas of machine learning and AI.