The Copycat Perceptron: Smashing Barriers Through Collective Learning

We characterize the equilibrium properties of a model of $y$ coupled binary perceptrons in the teacher-student scenario, subject to a suitable learning rule, with an explicit ferromagnetic coupling proportional to the Hamming distance between the students' weights. In contrast to recent works, we analyze a more general setting in which a thermal noise is present that affects the generalization performance of each student. Specifically, in the presence of a nonzero temperature, which assigns nonzero probability to configurations that misclassify samples with respect to the teacher's prescription, we find that the coupling of replicas leads to a shift of the phase diagram to smaller values of $\alpha$: This suggests that the free energy landscape gets smoother around the solution with good generalization (i.e., the teacher) at a fixed fraction of reviewed examples, which allows local update algorithms such as Simulated Annealing to reach the solution before the dynamics gets frozen. Finally, from a learning perspective, these results suggest that more students (in this case, with the same amount of data) are able to learn the same rule when coupled together with a smaller amount of data.