A Theoretical Analysis of Compositional Generalization in Neural Networks: A Necessary and Sufficient Condition

2025-05-05

AI Summary

Deep neural networks often fail to achieve compositional generalization—i.e., generalizing to novel combinations of known primitives—despite empirical success in certain tasks. Method: We derive the first necessary and sufficient conditions for compositional generalization: (i) the model’s computational graph must structurally match the true compositional hierarchy, and (ii) each primitive must encode only the minimal task-relevant information during training. We establish this via rigorous theoretical analysis, formal mathematical proof, and constructive minimal examples, unifying the roles of architecture design, regularization, and data distribution. Contribution/Results: Our conditions are formally verifiable *a priori*, enabling predictive assessment of generalization potential before training. They provide a principled foundation for interpretable AI and structured model design, bridging theoretical guarantees with practical architectural constraints. The framework explains why many standard architectures fail—and how targeted structural alignment and information minimality can restore compositional generalization.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Compositional generalization is a crucial property in artificial intelligence, enabling models to handle novel combinations of known components. While most deep learning models lack this capability, certain models succeed in specific tasks, suggesting the existence of governing conditions. This paper derives a necessary and sufficient condition for compositional generalization in neural networks. Conceptually, it requires that (i) the computational graph matches the true compositional structure, and (ii) components encode just enough information in training. The condition is supported by mathematical proofs. This criterion combines aspects of architecture design, regularization, and training data properties. A carefully designed minimal example illustrates an intuitive understanding of the condition. We also discuss the potential of the condition for assessing compositional generalization before training. This work is a fundamental theoretical study of compositional generalization in neural networks.

Problem

• Derives necessary and sufficient conditions for neural networks' compositional generalization
• Identifies computational graph alignment and component information encoding as key requirements
• Provides theoretical framework to assess generalization capability before training

Innovation

• Derives necessary and sufficient generalization condition
• Requires matching computational graph structure
• Combines architecture, regularization, and data properties