Hopfield model with planted patterns: a teacher-student self-supervised learning model

• Published in: Applied mathematics and computation 2023-12, Vol.458, p.128253, Article 128253
• Main Authors: Alemanno, Francesco, Camanzi, Luca, Manzan, Gianluca, Tantari, Daniele
• Format: Article
• Language: English
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2023.128253

While Hopfield networks are known as paradigmatic models for memory storage and retrieval, modern artificial intelligence systems mainly stand on the machine learning paradigm. We show that it is possible to formulate a teacher-student self-supervised learning problem with Boltzmann machines in terms of a suitable generalization of the Hopfield model with structured patterns, where the spin variables are the machine weights and patterns correspond to the training set's examples. We analyze the learning performance by studying the phase diagram in terms of the training set size, the dataset noise and the inference temperature (i.e. the weight regularization). With a small but informative dataset the machine can learn by memorization. With a noisy dataset, an extensive number of examples above a critical threshold is needed. In this regime the memory storage limits becomes an opportunity for the occurrence of a learning regime in which the system can generalize. •A teacher-student self-supervised learning problem with Boltzmann machines formulated as a Hopfield model with structured patterns.•Learning performance studied in terms of training set size, dataset noise and inference temperature.•With a small but informative dataset the machine can learn by memorization.•The memory storage limits becomes an opportunity for a learning regime in which the system can generalize.