A Shared Geometry of Difficulty in Multilingual Language Models
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arXiv
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| Format: | Preprint |
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2026
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| author | Civelli, Stefano Bernardelle, Pietro Brunello, Nicolò Demartini, Gianluca |
| author_facet | Civelli, Stefano Bernardelle, Pietro Brunello, Nicolò Demartini, Gianluca |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12731 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Shared Geometry of Difficulty in Multilingual Language Models Civelli, Stefano Bernardelle, Pietro Brunello, Nicolò Demartini, Gianluca Computation and Language Artificial Intelligence 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. |
| title | A Shared Geometry of Difficulty in Multilingual Language Models |
| topic | Computation and Language Artificial Intelligence |
| url | https://arxiv.org/abs/2601.12731 |