Quantifying Noise in Language Generation

Kleinberg and Mullainathan recently proposed a formal framework for studying the phenomenon of language generation, called language generation in the limit. In this model, an adversary gives an enumeration of example strings from an unknown target language, and the algorithm is tasked with correctly generating unseen strings from the target language within finite time. Refined notions of non-uniform and uniform generation were later introduced by Li, Raman, and Tewari (2025), and a noisy model was introduced by Raman and Raman (2025), which allows the adversary to insert extraneous strings. A natural question in the noisy model is to quantify the effect of noise, by studying the impact of each additional extraneous string. We show two complementary results in this setting. We first show that for both uniform and non-uniform generation, a single noisy string strictly reduces the set of collections that can be generated, thus answering an open question in Raman and Raman (2025). Then, we show for both uniform and non-uniform generation that generation with a single noisy string is equivalent to generation with any finite amount of noise, sharply contrasting with the strict hierarchy for noisy generation in the limit shown by Bai, Panigrahi, and Zhang (2026). Finally, we leverage our previous results to provide the first known characterization for non-uniform noise-dependent generatability.

💡 Research Summary

The paper investigates the impact of noisy training data on the theoretical capabilities of language generation models within the “language generation in the limit” framework originally introduced by Kleinberg and Mullainathan (2024). In this setting an adversary enumerates all strings of an unknown target language K and a learner must eventually output unseen strings from K after a finite time t*. Li, Raman, and Tewari (2025) refined this model by distinguishing uniform generation (where t* is independent of both the target language and the enumeration) and non‑uniform generation (where t* may depend on the language but not on the enumeration).

Raman and Raman (2025) extended the model to allow the adversary to insert a finite number n* of extraneous strings—noise—into the enumeration. The learner receives this noisy stream without knowing which strings are noise and must still generate correct unseen strings from K. They defined uniform and non‑uniform noise‑dependent generation (where the deadline t* may depend on the noise level) as well as noise‑independent variants (where t* must work for any noise level). Bai, Panigrahi, and Zhang (2026) showed that for the most permissive notion—noise‑independent generation—a strict hierarchy exists: for each i≥0 there is a collection generable with noise level i but not with i+1.

The present work focuses on the finer‑grained, noise‑dependent notions. Its first main result (Theorem 2.15) constructs a concrete collection that is generable without noise (hence both uniformly and non‑uniformly) but becomes non‑generable as soon as a single noisy string is introduced. This answers an open question from Raman and Raman in the strongest possible way: non‑uniform generation is not equivalent to non‑uniform noise‑dependent generation, even with just one noisy example.

The second main result (Theorem 2.14) establishes a surprising equivalence: for any finite noise level i≥1, a collection is uniformly (or non‑uniformly) generable with noise level i if and only if it is uniformly (or non‑uniformly) generable with noise level 1. In other words, the presence of a single noisy string collapses the entire hierarchy of finite noise levels; additional noise does not further restrict the learner’s power. This stands in stark contrast to the strict hierarchy for noisy generation in the limit proved by Bai et al.

To prove these statements the authors employ the “noisy closure” operator ⟨S⟩_{C,i} and the associated noisy‑closure dimension NC_i(C) introduced by Raman and Raman. The noisy closure of a set S captures the intersection of all languages in the collection C that are consistent with S given at most i noisy examples. Lemma 2.11 shows that any language consistent with S must contain the closure, so the learner can safely output only strings from this closure after seeing S. The authors show that non‑uniform noise‑dependent generation is possible exactly when every finite S has an infinite noisy closure, which matches the known characterization for non‑uniform generation without noise.

By combining the equivalence of noise levels with the closure characterization, the paper derives the first known structural description of collections that are non‑uniformly noise‑dependent generable. It also clarifies the relationship between uniform and non‑uniform notions: uniform noise‑dependent generation coincides with uniform generation at noise level 1, and similarly for the non‑uniform case.

Overall, the paper delivers three key insights: (1) a single noisy example can destroy the generability of collections that are otherwise easy to generate; (2) once a single noisy example is allowed, any finite amount of additional noise is irrelevant for both uniform and non‑uniform generation; (3) these phenomena enable a complete characterization of non‑uniform noise‑dependent generability. The results have practical relevance for large language models trained on imperfect data sources, suggesting that robustness to a few mislabeled examples may be fundamentally limited, yet beyond that point additional noise does not further degrade the model’s theoretical generation capacity.