Title: Universal Learning of Nonlinear Dynamics
Primary area: learning on time series and dynamical systems
Keywords: dynamical systems, online learning, spectral filtering

Abstract: We study the fundamental problem of learning a marginally stable unknown nonlinear dynamical system. We describe an algorithm for this problem, based on the technique of spectral filtering, which learns a mapping from past observations to the next based on a spectral representation of the system. Using techniques from online convex optimization, we prove vanishing prediction error for any nonlinear dynamical system that has finitely many marginally stable modes, with rates governed by a novel quantitative control-theoretic notion of learnability. The main technical component of our method is a new spectral filtering algorithm for linear dynamical systems, which incorporates past observations and applies to general noisy and marginally stable systems. This significantly generalizes the original spectral filtering algorithm to both asymmetric dynamics as well as incorporating noise correction, and is of independent interest.

TL;DR: We describe a prediction algorithm (spectral filtering) for nonlinear dynamical systems, and using linearization and improper learning we are able to prove online regret bounds for the next-token prediction task.