Order-Invariant Answers, Order-Sensitive Representations in Mathematical Reasoning

Problem

This work addresses the gap in understanding how answer invariance relates to representation invariance in mathematical reasoning tasks. The authors investigate this relationship through the lens of synthetic multi-step function-composition problems, which feature multiple rule orderings. The study is particularly relevant as it is presented as a preprint and has not undergone peer review.

Method

The authors utilize a dataset of synthetic multi-step function-composition problems, designed to test the models' reasoning capabilities across various rule orderings. They employ 16 language models with parameter sizes ranging from 1 billion to 8 billion. The primary metrics for evaluation include accuracy and permutation signal-to-noise ratio (SNR). To analyze the correlation between the models' performance and their internal representations, the authors compute Spearman correlations for layer-averaged permutation SNR and accuracy.

Results

The study reports a Spearman correlation of 0.86 between layer-averaged permutation SNR and accuracy, indicating a strong relationship between the two metrics. No other baselines or quantitative results are provided in the available text.

Limitations

The authors do not report any limitations in their study. However, the lack of peer review may imply that the findings should be interpreted with caution until validated by the community.

Why it matters

This research has implications for the design of language models in mathematical reasoning tasks, suggesting that understanding the relationship between answer and representation invariance could lead to improved model architectures and training methodologies. The findings may inform future work on enhancing model robustness and interpretability in complex reasoning scenarios.