Differentially Private Language Generation and Identification in the Limit

2026-04-09

AI Summary

This study investigates the feasibility and cost of language generation and recognition under differential privacy constraints. It introduces differential privacy into this setting for the first time, integrating extreme learning theory, computability analysis, and sample complexity lower bounds to enable effective task performance while preserving the privacy of input sequences in a continual release model. The core contributions include achieving private generation for countable language classes without qualitative loss, establishing the impossibility of private recognition for certain language structures, and revealing a necessary and sufficient condition under which private recognition in stochastic settings is equivalent to that in adversarial settings. These results elucidate the fundamental disparity between generation and recognition tasks in terms of their inherent privacy costs.

Abstract

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.

Problem

• differential privacy
• language generation
• language identification
• in the limit
• privacy constraints

Innovation

• differential privacy
• language generation in the limit
• language identification
• sample complexity
• adversarial vs stochastic settings