Thermodynamic limit in learning period three

TL;DR

This paper investigates how neural networks trained on three data points can generate a wide variety of periodic orbits, revealing universal bifurcation behaviors and properties of learned dynamics in the thermodynamic limit.

Contribution

It demonstrates that neural networks can produce all periodic orbits from three data points and identifies a universal bifurcation scenario linked to the logistic map.

Findings

Almost all learned periods are unstable.

A universal bifurcation scenario appears in quadratic interpolation.

Networks exhibit specific properties like finite-size effects and symmetry in learning period three.

Abstract

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.