Learning Sparse Nonparametric DAGs (1909.13189v2)

Abstract: We develop a framework for learning sparse nonparametric directed acyclic graphs (DAGs) from data. Our approach is based on a recent algebraic characterization of DAGs that led to a fully continuous program for score-based learning of DAG models parametrized by a linear structural equation model (SEM). We extend this algebraic characterization to nonparametric SEM by leveraging nonparametric sparsity based on partial derivatives, resulting in a continuous optimization problem that can be applied to a variety of nonparametric and semiparametric models including GLMs, additive noise models, and index models as special cases. Unlike existing approaches that require specific modeling choices, loss functions, or algorithms, we present a completely general framework that can be applied to general nonlinear models (e.g. without additive noise), general differentiable loss functions, and generic black-box optimization routines. The code is available at https://github.com/xunzheng/notears.

• The paper introduces a continuous optimization framework that unifies nonlinear and nonparametric structural equation models for sparse DAG learning.
• It extends smooth acyclicity characterization from linear setups to complex models, enabling detection of nonlinear dependencies.
• Empirical evaluations demonstrate improved accuracy and efficiency over state-of-the-art methods using global edge updates and versatile estimators.

Learning Sparse Nonparametric DAGs: An Overview

This paper addresses the challenge of learning sparse nonparametric directed acyclic graphs (DAGs), an enduring problem in machine learning with broad applications such as causal inference, medicine, and finance. Existing methodologies often depend on specific model assumptions or algorithms, necessitating expert knowledge for model and algorithm selection. This work introduces a general framework for learning DAG models, incorporating nonparametric approaches, thus overcoming the limitations of previous, more restricted methods.

Framework and Methodology

The authors develop a framework leveraging a recent algebraic characterization of DAGs, initially targeted at linear structural equation models (SEMs), extending it to nonparametric SEMs. By using nonparametric sparsity characterized by partial derivatives, the authors reformulate the problem into a continuous optimization framework applicable to a wide array of models, including generalized linear models (GLMs), additive noise, and index models.

This general framework enables the learning of DAGs without committing to specific modeling choices or methods. It integrates well with general nonlinear models and differentiable loss functions, engaging generic black-box optimization routines.

Key Contributions

The paper's primary contributions are fourfold:

1. Generic Optimization Problem: The authors introduce a continuous optimization problem that captures nonlinear and nonparametric SEMs, addressing a range of models without needing specialized algorithms.
2. Acyclicity Characterization Extension: Building on previous work, they extend the smooth characterization of acyclicity to nonparametric models, showing that the linear parametrization is merely a special case. They apply this to several nonlinear dependencies.
3. Nonparametric Estimators: Two classes of nonparametric estimators are explored: neural networks and orthogonal basis expansions. The paper explores their properties and their role within this framework.
4. Empirical Validation: Through comprehensive empirical evaluation against contemporary state-of-the-art approaches, the framework's versatility and efficacy are demonstrated across diverse models.

Implications and Future Directions

Practically, this framework suggests a method for simultaneous edge updates using global information over the DAG—contrasting with local search methods—and thus may lead to more efficient algorithms in terms of computation and performance. Theoretically, it opens up pathways for employing robust optimizers from the optimization literature, potentially integrating more sophisticated techniques in the future.

Empirical validation through simulations and real data showcases the framework's competitive performance, often surpassing existing methods in accuracy and runtime efficiency. The method's capacity to generalize across different network structures and dependencies, emphasized by its broader applicability compared to kernel-based methods, positions it as a promising direction in the pursuit of flexible and scalable DAG learning solutions.

The paper has laid foundational work that not only advances the field of DAG learning but also sets the stage for further exploration into sophisticated nonparametric estimations and optimization strategies. Its potential for refinement and adaptation could significantly influence theoretical and applied machine learning research, particularly in fields prioritizing interpretability and causal link structures.

Conclusion

In summary, this paper's contribution lies in its unifying framework for sparse nonparametric DAG learning, bridging gaps left by specialist methods while maintaining high adaptability and strong empirical results. Its robustness in applying generalized solutions to diverse scenarios holds promise for ongoing advancements in machine learning.