Beta Embeddings for Multi-Hop Logical Reasoning in Knowledge Graphs (2010.11465v1)

Abstract: One of the fundamental problems in Artificial Intelligence is to perform complex multi-hop logical reasoning over the facts captured by a knowledge graph (KG). This problem is challenging, because KGs can be massive and incomplete. Recent approaches embed KG entities in a low dimensional space and then use these embeddings to find the answer entities. However, it has been an outstanding challenge of how to handle arbitrary first-order logic (FOL) queries as present methods are limited to only a subset of FOL operators. In particular, the negation operator is not supported. An additional limitation of present methods is also that they cannot naturally model uncertainty. Here, we present BetaE, a probabilistic embedding framework for answering arbitrary FOL queries over KGs. BetaE is the first method that can handle a complete set of first-order logical operations: conjunction ($\wedge$), disjunction ($\vee$), and negation ($\neg$). A key insight of BetaE is to use probabilistic distributions with bounded support, specifically the Beta distribution, and embed queries/entities as distributions, which as a consequence allows us to also faithfully model uncertainty. Logical operations are performed in the embedding space by neural operators over the probabilistic embeddings. We demonstrate the performance of BetaE on answering arbitrary FOL queries on three large, incomplete KGs. While being more general, BetaE also increases relative performance by up to 25.4% over the current state-of-the-art KG reasoning methods that can only handle conjunctive queries without negation.

• The paper introduces Beta Embeddings, modeling entities and queries as Beta distributions to perform complete first-order logical operations including negation.
• It leverages probabilistic projection, intersection, and negation operators to achieve up to 25.4% performance improvement on standard benchmarks like FB15k and NELL995.
• The framework robustly handles uncertainty in knowledge graphs, offering enhanced reasoning capabilities for AI systems and paving the way for future probabilistic research.

Beta Embeddings for Multi-Hop Logical Reasoning in Knowledge Graphs

The paper "Beta Embeddings for Multi-Hop Logical Reasoning in Knowledge Graphs" addresses the challenge of performing complex logical reasoning over large-scale, incomplete knowledge graphs (KGs). The authors introduce a novel probabilistic embedding framework capable of handling arbitrary first-order logic (FOL) queries, including negation, which is notably absent from previous approaches.

Key Contributions

The primary contribution of the paper is the introduction of a method, denoted as Beta Embeddings, which models both KG entities and queries as probabilistic distributions. This innovative approach facilitates the execution of complete FOL operations—conjunction, disjunction, and importantly, negation—directly in the embedding space. The probabilistic nature of the embeddings also allows for a coherent modeling of uncertainty, an essential aspect of reasoning with real-world, incomplete KGs.

Methodology

The proposed framework leverages the Beta distribution, defined over the bounded interval [0,1], to represent entities and queries. This choice of distribution is instrumental in defining the probabilistic logical operators due to its properties:

1. Probabilistic Projection: Implements transformations for relation types using multi-layer perceptrons, allowing entities to transition across the graph based on specific relations.
2. Probabilistic Intersection: Uses a weighted product of probability density functions to intersect multiple distributions. This operator is computationally efficient and remains within the bounds of feasible solution constructs.
3. Probabilistic Negation: Achieves negation by transforming regions of high probability density to low and vice versa, by inverting the parameters of the Beta distribution. This enables the method to support a complete set of logical operations in FOL.

The model’s ability to handle disjunctive operations via De Morgan's laws further illustrates its robustness. However, the embedding dimensionality requirement poses a limit when representing unions exactly, a well-recognized issue in the theoretical space. Empirically, high-dimensional embeddings mitigate this limitation adequately in practical scenarios. The approach remains linear in complexity with respect to the query operators.

Evaluation

The efficacy of the method was tested on three standard benchmark datasets: FB15k, FB15k-237, and NELL995, encompassing a wide range of query structures. The results indicated an impressive relative performance improvement of up to 25.4% over previous state-of-the-art methods for conjunctive queries. Additionally, the methodology generalized well across query structures it was not explicitly trained on, particularly those involving negation, thus validating the framework's capacity to handle complex real-world reasoning tasks.

Implications and Future Directions

The introduction of a probabilistic approach for logical reasoning marks a significant paradigm shift in the way knowledge graph queries can be interpreted and answered. This model not only supports more comprehensive logical expressions but also provides a viable path toward handling uncertainty within KGs, a feature lacking in earlier deterministic models.

Practically, such a framework can improve the quality and scope of AI systems reliant on KG reasoning, such as advanced search engines, recommendation systems, and automated knowledge synthesis tools. Theoretically, it opens avenues for further research into probabilistic embeddings and the potential for integrating such structures with symbolic reasoning methods. Future work could explore optimization of embedding dimensionality, improved handling of edge-case queries, or integration with other probabilistic models to enhance interpretability and reasoning power further.

In conclusion, "Beta Embeddings for Multi-Hop Logical Reasoning in Knowledge Graphs" presents a substantial advancement in KG reasoning by accommodating full FOL capabilities and offering robust empirical performance, setting a new standard for future research in this domain.