Convergent Evolution: How Different Language Models Learn Similar Number Representations

TL;DR

This paper investigates how various language models learn number representations, revealing a hierarchy of features and the conditions under which models develop geometrically separable features, demonstrating convergent evolution.

Contribution

It identifies the conditions and training signals that lead different models to learn similar number features, and provides theoretical insights into Fourier domain sparsity and geometric separability.

Findings

Models learn periodic number features with dominant Fourier periods.

Fourier sparsity is necessary but not sufficient for geometric separability.

Training data, architecture, optimizer, and tokenizer influence feature learning.

Abstract

Language models trained on natural text learn to represent numbers using periodic features with dominant periods at $T=2, 5, 10$. In this paper, we identify a two-tiered hierarchy of these features: while Transformers, Linear RNNs, LSTMs, and classical word embeddings trained in different ways all learn features that have period-$T$ spikes in the Fourier domain, only some learn geometrically separable features that can be used to linearly classify a number mod-$T$. To explain this incongruity, we prove that Fourier domain sparsity is necessary but not sufficient for mod-$T$ geometric separability. Empirically, we investigate when model training yields geometrically separable features, finding that the data, architecture, optimizer, and tokenizer all play key roles. In particular, we identify two different routes through which models can acquire geometrically separable features: they can learn them from complementary co-occurrence signals in general language data, including text-number co-occurrence and cross-number interaction, or from multi-token (but not single-token) addition problems. Overall, our results highlight the phenomenon of convergent evolution in feature learning: A diverse range of models learn similar features from different training signals.