A computational phase transition for learning-to-sample from Ising models

Authors: Andrej Risteski, Thuy-Duong Vuong

We study \emph{learning-to-sample} -- a basic algorithmic task underlying generative modeling -- for Ising models, a standard testbed for algorithmic ideas in both theoretical computer science and machine learning. Given i.i.d. samples of an unknown target distribution, the goal of learning-to-sample is to learn a computationally efficient generation procedure that produces new samples following approximately the same distribution. We construct a family of Ising models of constantly bounded-width which lie just beyond the spectral threshold $λ_{\max}(J)-λ_{\min}(J)=1$, and show that learning-to-sample for this family is computationally hard under standard cryptographic assumptions, even when the learner is given both polynomially many i.i.d. samples from the model and explicit access to its parameters. Combined with results of [AJKPV24,KLV25] showing tractability of learning-to-sample below the spectral threshold, this establishes a sharp computational phase transition at the spectral threshold. Moreover, combined with prior results on parameter learning for bounded-width Ising models [KM17,WSD19,VML20], this shows that learning-to-sample can be more difficult than parameter learning. Finally, we show that any efficient learner for these hard instances exhibits a natural memorization-hallucination dichotomy: the learner must either output configurations that, after a simple transformation, match the (transformed) training data or place substantial mass on configurations of negligible probability under the target distribution.