Dense Hopfield Networks in the Teacher-Student Setting

Dense Hopfield networks are known for their feature to prototype transition and adversarial robustness. However, previous theoretical studies have been mostly concerned with their storage capacity. We bridge this gap by studying the phase diagram of p-body Hopfield networks in the teacher-student setting of an unsupervised learning problem, uncovering ferromagnetic phases reminiscent of the prototype and feature learning regimes. On the Nishimori line, we find the critical size of the training set necessary for efficient pattern retrieval. Interestingly, we find that that the paramagnetic to ferromagnetic transition of the teacher-student setting coincides with the paramagnetic to spin-glass transition of the direct model, i.e. with random patterns. Outside of the Nishimori line, we investigate the learning performance in relation to the inference temperature and dataset noise. Moreover, we show that using a larger p for the student than the teacher gives the student an extensive tolerance to noise. We then derive a closed-form expression measuring the adversarial robustness of such a student at zero temperature, corroborating the positive correlation between number of parameters and robustness observed in large neural networks. We also use our model to clarify why the prototype phase of modern Hopfield networks is adversarially robust.