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Concept Generalization in Humans and Large Language Models: Insights from the Number Game

Arghavan Bazigaran, Hansem Sohn

TL;DR

The paper investigates how humans and LLMs generalize mathematical concepts using the number game, adopting a Bayesian framework as a normative benchmark. It contrasts human flexibility in combining rule-based and similarity-based concepts with GPT’s apparent emphasis on rule-based inferences, revealing poorer one-shot generalization and greater reliance on multiple examples for LLMs. By fitting a Bayesian model and testing variants (BinL, MAP, MaxL), the study shows humans align closely with Bayesian predictions, while GPT’s generalization is better captured by likelihood-driven, MaxL-like strategies and a higher rule bias. These findings illuminate fundamental differences in mathematical reasoning between humans and LLMs and suggest directions for improving LLM inductive biases through targeted training and prompting to foster richer numerical concept representations.

Abstract

We compare human and large language model (LLM) generalization in the number game, a concept inference task. Using a Bayesian model as an analytical framework, we examined the inductive biases and inference strategies of humans and LLMs. The Bayesian model captured human behavior better than LLMs in that humans flexibly infer rule-based and similarity-based concepts, whereas LLMs rely more on mathematical rules. Humans also demonstrated a few-shot generalization, even from a single example, while LLMs required more samples to generalize. These contrasts highlight the fundamental differences in how humans and LLMs infer and generalize mathematical concepts.

Concept Generalization in Humans and Large Language Models: Insights from the Number Game

TL;DR

The paper investigates how humans and LLMs generalize mathematical concepts using the number game, adopting a Bayesian framework as a normative benchmark. It contrasts human flexibility in combining rule-based and similarity-based concepts with GPT’s apparent emphasis on rule-based inferences, revealing poorer one-shot generalization and greater reliance on multiple examples for LLMs. By fitting a Bayesian model and testing variants (BinL, MAP, MaxL), the study shows humans align closely with Bayesian predictions, while GPT’s generalization is better captured by likelihood-driven, MaxL-like strategies and a higher rule bias. These findings illuminate fundamental differences in mathematical reasoning between humans and LLMs and suggest directions for improving LLM inductive biases through targeted training and prompting to foster richer numerical concept representations.

Abstract

We compare human and large language model (LLM) generalization in the number game, a concept inference task. Using a Bayesian model as an analytical framework, we examined the inductive biases and inference strategies of humans and LLMs. The Bayesian model captured human behavior better than LLMs in that humans flexibly infer rule-based and similarity-based concepts, whereas LLMs rely more on mathematical rules. Humans also demonstrated a few-shot generalization, even from a single example, while LLMs required more samples to generalize. These contrasts highlight the fundamental differences in how humans and LLMs infer and generalize mathematical concepts.
Paper Structure (21 sections, 5 equations, 6 figures, 1 table)

This paper contains 21 sections, 5 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods
  3. Human and LLM Dataset
  4. Bayesian Model
  5. Model Fitting and Evaluation
  6. Results
  7. Qualitative comparisons: no one-shot generalization in GPT
  8. Quantitative comparisons: baseline Bayes captures human behavior better than GPT
  9. Set-length effect: Poor sample efficiency in GPT
  10. Bayesian model fitting: higher rule bias in GPT
  11. Nested model comparison: maximum likelihood inference in GPT
  12. Other LLMs in the number game: less structured behavior
  13. Discussion
  14. Dataset Structure
  15. Bayesian Model Details
  16. ...and 6 more sections

Figures (6)

  • Figure 1: Task structure used for LLM evaluation. For each example set, the model is queried on all possible targets, and this process is repeated across trials.
  • Figure 2: Number-game behavior of human, GPT, and baseline Bayesian model (Bayes) for selected example sets (top titles), showing the probability that a target number $y$ belongs to the same concept $h$ as the given example set $X$. Part of the data consistent with a mathematical rule is shown in pale blue.
  • Figure 3: (A) Quantitative comparison of humans, GPT, and baseline Bayes. Mean divergence for each set length is shown for human–Bayes–GPT comparisons. (B) The central plot shows the per-set divergence: GPT-Bayes (y-axis) and Human-Bayes (x-axis). Marginal histograms show the value distributions. Error bars show the standard deviation. (C) Summary of the Bayesian models. The baseline, best fit and variants; BinL (binary likelihood), MAP (maximum a posteriori), and MaxL (maximum likelihood).
  • Figure B.1: Grid search optimization for $\alpha$ and $\lambda$ parameters. The red star marks the best parameter configuration. Below each panel, histograms show the distribution of per-set JSD values for the best fit.
  • Figure C.1: Per-set JSD comparisons between each variant (x-axis) and the baseline model (y-axis). Error bars show the standard deviation, clipped at the JSD bounds.
  • ...and 1 more figures