Differentially Private Language Generation and Identification in the Limit

We initiate the study of language generation in the limit, a model recently introduced by Kleinberg and Mullainathan [KM24], under the constraint of differential privacy. We consider the continual release model, where a generator must eventually output a stream of valid strings while protecting the privacy of the entire input sequence. Our first main result is that for countable collections of languages, privacy comes at no qualitative cost: we provide an $\varepsilon$-differentially-private algorithm that generates in the limit from any countable collection. This stands in contrast to many learning settings where privacy renders learnability impossible. However, privacy does impose a quantitative cost: there are finite collections of size $k$ for which uniform private generation requires $Ω(k/\varepsilon)$ samples, whereas just one sample suffices non-privately. We then turn to the harder problem of language identification in the limit. Here, we show that privacy creates fundamental barriers. We prove that no $\varepsilon$-DP algorithm can identify a collection containing two languages with an infinite intersection and a finite set difference, a condition far stronger than the classical non-private characterization of identification. Next, we turn to the stochastic setting where the sample strings are sampled i.i.d. from a distribution (instead of being generated by an adversary). Here, we show that private identification is possible if and only if the collection is identifiable in the adversarial model. Together, our results establish new dimensions along which generation and identification differ and, for identification, a separation between adversarial and stochastic settings induced by privacy constraints.

Language Generation with Infinite Contamination

We study language generation in the limit, where an algorithm observes an adversarial enumeration of strings from an unknown target language $K$ and must eventually generate new, unseen strings from $K$. Kleinberg and Mullainathan [KM24] proved that generation is achievable in surprisingly general settings. But their generator suffers from ``mode collapse,'' producing from an ever-smaller subset of the target. To address this, Kleinberg and Wei [KW25] require the generator's output to be ``dense'' in the target language. They showed that generation with density, surprisingly, remains achievable at the same generality. Both results assume perfect data: no noisy insertions and no omissions. This raises a central question: how much contamination can generation tolerate? Recent works made partial progress on this question by studying (non-dense) generation with either finite amounts of noise (but no omissions) or omissions (but no noise). We characterize robustness under contaminated enumerations: 1. Generation under Contamination: Language generation in the limit is achievable for all countable collections iff the fraction of contaminated examples converges to zero. When this fails, we characterize which collections are generable. 2. Dense Generation under Contamination: Dense generation is strictly less robust to contamination than generation. As a byproduct, we resolve an open question of Raman and Raman [ICML25] by showing that generation is possible with only membership oracle access under finitely many contaminated examples. Finally, we introduce a beyond-worst-case model inspired by curriculum learning and prove that dense generation is achievable even with infinite contamination provided the fraction of contaminated examples converges to zero. This suggests curriculum learning may be crucial for learning from noisy web data.