Title: Learning Proportional Analogies: Lightweight Neural Network vs LLM
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
Keywords: analogy, analogical reasoning, learnability, reasoning, proportional analogy

Abstract: Analogical reasoning often involves statements of the form $\alpha:\beta::\gamma:\delta$, known as proportional analogies, which can be interpreted as ''$\alpha$ differs from $\beta$ as  $\gamma$ differs from $\delta$''  and ''$\beta$ differs from $\alpha$ as $\delta$ differs from $\gamma$''. In this paper, we study the learnability of proportional analogies from both theoretical and experimental perspectives. We show that, in the Boolean setting—where each element of a proportional analogy is represented as a Boolean vector—proportional analogies are efficiently PAC learnable. To validate this in practice, we instantiate proportional analogies in a perceptual scenario with 4-cell images, each cell containing a shape and a color. We automatically generate a dataset of valid and invalid proportional analogies and train lightweight artificial neural networks (ANNs) as evaluators. We compare our ANN-based models against state-of-the-art Large Language Models (LLMs) in proportional analogy verification (checking correctness), proportional analogy generation (producing missing elements), and proportional analogy generalization (applying knowledge acquired during learning to unseen features). Our results show that lightweight ANNs i) match LLMs in verification and generalization, and ii) outperform LLMs in generation, demonstrating that simple, efficient models can effectively learn and generalize proportional analogies while using far fewer resources.

TL;DR: We discuss the learnability issues of proportional analogies, especially in the boolean setting. We show that simple ANN can rival state-of-the-art LLM's in learning proportional analogies regarding our experimental settings.