Thermodynamic limit in learning period three
This addresses a fundamental question in dynamical systems and machine learning about the minimal data needed for complex behavior, though it appears incremental as it builds on known concepts like the logistic map.
The paper tackles whether learning just three data points can enable a neural network to produce any periodic orbit, and finds that in the thermodynamic limit, networks can indeed develop attractors of all periods, with a universal bifurcation scenario emerging for quadratic interpolation.
A continuous one-dimensional map with period three includes all periods. This raises the following question: Can we obtain any types of periodic orbits solely by learning three data points? In this paper, we report the answer to be yes. Considering a random neural network in its thermodynamic limit, we first show that almost all learned periods are unstable, and each network has its own characteristic attractors (which can even be untrained ones). The latently acquired dynamics, which are unstable within the trained network, serve as a foundation for the diversity of characteristic attractors and may even lead to the emergence of attractors of all periods after learning. When the neural network interpolation is quadratic, a universal post-learning bifurcation scenario appears, which is consistent with a topological conjugacy between the trained network and the classical logistic map. In addition to universality, we explore specific properties of certain networks, including the singular behavior of the scale of weight at the infinite limit, the finite-size effects, and the symmetry in learning period three.