Generalizing Analogical Inference from Boolean to Continuous Domains

2025-11-13

AI Summary

Existing analogical reasoning frameworks are restricted to the Boolean domain and lack theoretical foundations for continuous domains and regression tasks. Method: We propose the first unified real-valued analogical reasoning theory, based on a parametric analogical model constructed via generalized means. This model characterizes, for the first time, the structural properties of analogy-preserving functions over continuous domains and derives tight worst-case and average-case error bounds under function smoothness assumptions. Contribution/Results: The framework seamlessly unifies Boolean classification and continuous regression, rectifying the inadequacy of classical generalization bounds in real-valued settings. It establishes the first provably guaranteed theoretical foundation for analogical reasoning in continuous domains, enabling rigorous performance analysis and principled extension to regression.

Technology Category

• Cognitive Modeling & Cognitive Systems: Analogy
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Analogical reasoning is a powerful inductive mechanism, widely used in human cognition and increasingly applied in artificial intelligence. Formal frameworks for analogical inference have been developed for Boolean domains, where inference is provably sound for affine functions and approximately correct for functions close to affine. These results have informed the design of analogy-based classifiers. However, they do not extend to regression tasks or continuous domains. In this paper, we revisit analogical inference from a foundational perspective. We first present a counterexample showing that existing generalization bounds fail even in the Boolean setting. We then introduce a unified framework for analogical reasoning in real-valued domains based on parameterized analogies defined via generalized means. This model subsumes both Boolean classification and regression, and supports analogical inference over continuous functions. We characterize the class of analogy-preserving functions in this setting and derive both worst-case and average-case error bounds under smoothness assumptions. Our results offer a general theory of analogical inference across discrete and continuous domains.

Problem

• Extends analogical inference from Boolean to continuous domains for regression
• Develops unified framework using generalized means for real-valued analogies
• Characterizes analogy-preserving functions with error bounds across domains

Innovation

• Unified analogical reasoning framework using parameterized analogies
• Generalized means enable continuous domain analogical inference
• Characterizes analogy-preserving functions with error bounds