Higher Embedding Dimension Creates a Stronger World Model for a Simple Sorting Task

TL;DR

This study shows that increasing the embedding dimension in transformer models enhances the internal world model's structure, interpretability, and robustness in a simple sorting task, despite high accuracy being achievable with small dimensions.

Contribution

It provides empirical evidence that larger embedding dimensions lead to more faithful and interpretable internal representations in transformer models performing sorting tasks.

Findings

Higher embedding dimensions improve internal representation quality.

Attention weights encode global token order monotonically.

Transpositions align with largest adjacent differences.

Abstract

We investigate how embedding dimension affects the emergence of an internal "world model" in a transformer trained with reinforcement learning to perform bubble-sort-style adjacent swaps. Models achieve high accuracy even with very small embedding dimensions, but larger dimensions yield more faithful, consistent, and robust internal representations. In particular, higher embedding dimensions strengthen the formation of structured internal representation and lead to better interpretability. After hundreds of experiments, we observe two consistent mechanisms: (1) the last row of the attention weight matrix monotonically encodes the global ordering of tokens; and (2) the selected transposition aligns with the largest adjacent difference of these encoded values. Our results provide quantitative evidence that transformers build structured internal world models and that model size improves representation quality in addition to end performance. We release our metrics and analyses, which can be used to probe similar algorithmic tasks.