Newest papers

A Reverse-Gold Theorem: Positive-Data Identification by Substrate-Bounded Contrastive Learners

Rakesh Sengupta · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.28747 · cited by 0 works (OpenAlex) · open-access PDF

French

Le théorème du contexte de l'opérateur open-key de lifting de Gold montre qu'aucun apprenant ne peut identifier un langage arbitraire dans une classe superfinite à partir de présentations positives seules. Nous prouvons un résultat positif d'une forme différente. Fixez un opérateur de contraste récursif inné \emph{$R$ mappant toute chaîne à un ensemble fini de perturbations structurellement limitées de celui-ci, et un scorer discriminant d'identification conditionnelle de capacité limitée formé par des données positives seules: avec une probabilité élevée, il se classe tout sauf une estimation de contraste \citep{vandenoord2018,macollins2018} sur les perturbations de contraste auto-générées. So

Machine translation (Meta M2M100); the English original follows.

Abstract

Gold's theorem shows that no learner can identify an arbitrary language in a superfinite class from positive presentations alone. We prove a positive result of a different shape. Fix an innate \emph{recursive contrast operator} $R$ mapping any string to a finite set of structurally bounded perturbations of it, and a discriminative scorer of bounded capacity trained by InfoNCE-style contrastive estimation \citep{vandenoord2018,macollins2018} on self-generated perturbations. Under a faithfulness condition relating operator, scorer, and target, the trained scorer achieves \emph{PAC conditional identification} from positive data alone: with high probability it ranks all but an $\varepsilon$-fraction of positives above every out-of-language perturbation in their own orbit, with sample complexity polynomial in the capacity, the orbit size, the score bound, and $1/\varepsilon$. The negatives are constructed by the learner, not supplied by an oracle, and no global normalizing constant over strings is ever computed. The misranking bound follows from the geometry of the contrastive loss---per-example loss floors plus a definite loss increment under misranking---yielding an $\varepsilon^{-2}$ rate, together with an explicit approximation gap that quantifies the cost of in-language (``colliding'') perturbations and vanishes for orbit-pure operators. The ranking guarantee lifts to a global membership classifier under a margin condition, and the same operator composed with the learned scorer is a normalization-free Metropolis--Hastings generator. We instantiate the hypotheses for substitutable context-free languages \citep{clarkeyraud2007} and the subregular classes \citep{heinz2010,hayashikamigaito2025}, and state an open conjecture locating the construction in the mildly context-sensitive hierarchy.

Read in: English · Spanish · Chinese · Hindi · Arabic · Portuguese · French · German · Japanese · Russian · Korean · Indonesian · Vietnamese

Topics: Bounded function, Discriminative model, Classifier (UML), Mathematics, Computer science, Conjecture

Sources: OpenAlex (CC0), arXiv, Crossref. Agent Exchange