A Shared Geometry of Difficulty in Multilingual Language Models

TL;DR

This paper investigates how large multilingual language models internally represent problem difficulty, revealing a two-stage process where initial language-agnostic signals become language-specific, with implications for understanding model interpretability.

Contribution

It uncovers a two-stage internal representation of difficulty in multilingual LLMs, showing how early signals are language-agnostic and later signals become language-specific.

Findings

Deep representations excel in within-language difficulty prediction.

Shallow representations generalize better across languages.

Difficulty signals emerge at different internal layers of the model.

Abstract

Predicting problem-difficulty in large language models (LLMs) refers to estimating how difficult a task is according to the model itself, typically by training linear probes on its internal representations. In this work, we study the multilingual geometry of problem-difficulty in LLMs by training linear probes using the AMC subset of the Easy2Hard benchmark, translated into 21 languages. We found that difficulty-related signals emerge at two distinct stages of the model internals, corresponding to shallow (early-layers) and deep (later-layers) internal representations, that exhibit functionally different behaviors. Probes trained on deep representations achieve high accuracy when evaluated on the same language but exhibit poor cross-lingual generalization. In contrast, probes trained on shallow representations generalize substantially better across languages, despite achieving lower within-language performance. Together, these results suggest that LLMs first form a language-agnostic representation of problem difficulty, which subsequently becomes language-specific. This closely aligns with existing findings in LLM interpretability showing that models tend to operate in an abstract conceptual space before producing language-specific outputs. We demonstrate that this two-stage representational process extends beyond semantic content to high-level meta-cognitive properties such as problem-difficulty estimation.