Generative Flow Networks: Theory and Applications to Structure Learning (2501.05498v1)

Abstract: Without any assumptions about data generation, multiple causal models may explain our observations equally well. To avoid selecting a single arbitrary model that could result in unsafe decisions if it does not match reality, it is therefore essential to maintain a notion of epistemic uncertainty about our possible candidates. This thesis studies the problem of structure learning from a Bayesian perspective, approximating the posterior distribution over the structure of a causal model, represented as a directed acyclic graph (DAG), given data. It introduces Generative Flow Networks (GFlowNets), a novel class of probabilistic models designed for modeling distributions over discrete and compositional objects such as graphs. They treat generation as a sequential decision making problem, constructing samples of a target distribution defined up to a normalization constant piece by piece. In the first part of this thesis, we present the mathematical foundations of GFlowNets, their connections to existing domains of machine learning and statistics such as variational inference and reinforcement learning, and their extensions beyond discrete problems. In the second part of this thesis, we show how GFlowNets can approximate the posterior distribution over DAG structures of causal Bayesian Networks, along with the parameters of its causal mechanisms, given observational and experimental data.

• The paper introduces GFlowNets as a novel probabilistic framework that navigates complex discrete structures using flow matching conditions to ensure valid trajectory flows.
• It applies these models to Bayesian network structure learning, efficiently approximating posterior distributions while overcoming biases inherent in traditional reinforcement learning.
• Empirical results demonstrate that GFlowNets significantly enhance inference efficiency and sample fidelity, paving the way for more robust probabilistic modeling techniques.

Generative Flow Networks: Theory and Applications to Structure Learning

The academic work under discussion is centered on Generative Flow Networks (GFlowNets) and their theoretical and practical applications in learning structures, such as Bayesian networks. This discourse is pivotal in the domain of computational sciences, especially where the necessity arises to efficiently sample from distributions defined over intricate discrete structures like graphs without explicitly computing normalization constants, a task that remains computationally infeasible due to its combinatorial nature.

Overview of GFlowNets

GFlowNets essentially introduce an innovative class of probabilistic models designed to navigate and sample from distributions over discrete and compositional structures, such as directed acyclic graphs (DAGs). These models are pertinent in areas where existing inference techniques falter due to intractability, such as Bayesian statistics involving large hypothesis spaces. The intrinsic merit of GFlowNets lies in their ability to approximate posterior distributions over complex structures by treating each potential model as a possible trajectory through a network.

Theoretical Insights

The paper offers significant theoretical insights into how GFlowNets can be used to model distributions by leveraging flow networks. It delineates multiple methods to ascertain valid trajectory flows through the application of "flow matching conditions," which ensure that the flow within a network matches the predefined constraints dictated by the structure learning problem. The robustness of GFlowNets is illustrated by their mathematical foundation where flow conservation laws akin to those in fluid dynamics are employed.

Through these networks, a flow assigned to DAG elements corresponds to their relative probability under a target distribution, up to a normalization constant. This approach effectively overcomes the biases introduced by reinforcement learning strategies, particularly where multiple paths can generate the same structure, which is not accounted for in traditional RL frameworks. In this light, GFlowNets are presented as a versatile framework for accurate sampling without falling prey to the inherent inductive biases of models trained traditionally through RL techniques.

Practical Applications

The paper extends the implications of theoretical developments in GFlowNets to practical applications, notably in the structure learning of Bayesian networks. In this context, the task is to learn both the architecture and parameterized relationships within networks from observational data—a task compounded by issues of identifiability due to latent equivalence classes of structures. GFlowNets aid in this regard by enabling a Bayesian approach to structure learning, offering a probabilistic methodology that does not resort to arbitrary selections among equivalent structures but instead refines posterior distributions through sequential decision processes.

Numerical Outcomes and Innovative Directions

From a numerical viewpoint, GFlowNets have shown to significantly improve inference efficiency while maintaining high fidelity to the target distribution, even when faced with large and sparse DAGs. The studies at hand illustrate the effectiveness of GFlowNets in empirical settings, highlighting their capacity for integrative learning of both the structure and parameters of Bayesian networks to jointly represent their posterior distributions.

These findings clear a path for innovative directions in AI, where GFlowNets could redefine standard practices for model selection and hypothesis testing in statistics and machine learning. Their potential for extensions into continuous and high-dimensional spaces also presents a frontier for future exploration, allowing them to serve as templates for solving a broader array of probabilistic inference and generative modeling challenges.

Conclusion

Ultimately, the incorporation of Generative Flow Networks into the repertoire of structured probabilistic modeling tools marks a pivotal enhancement in sampling-based inference. They promise a new horizon for how structures are learned and inferred, presenting an approach that could potentially disrupt conventional methodologies by providing a more nuanced, refined, and tractable resolution of complex probabilistic graph modeling tasks.