Research Program: Theory of Learning in Dynamical Systems

TL;DR

This paper proposes a framework for understanding the learnability of dynamical systems from observations, emphasizing finite-sample guarantees and properties like stability and spectral features, with applications to linear systems.

Contribution

It introduces a new notion of dynamic learnability based on system properties, and demonstrates finite-sample prediction guarantees without system identification for linear systems.

Findings

Finite-sample guarantees for learning in dynamical systems

Spectral filtering enables accurate prediction without system identification

Framework connects learnability with system stability and spectral properties

Abstract

Modern learning systems increasingly interact with data that evolve over time and depend on hidden internal state. We ask a basic question: when is such a dynamical system learnable from observations alone? This paper proposes a research program for understanding learnability in dynamical systems through the lens of next-token prediction. We argue that learnability in dynamical systems should be studied as a finite-sample question, and be based on the properties of the underlying dynamics rather than the statistical properties of the resulting sequence. To this end, we give a formulation of learnability for stochastic processes induced by dynamical systems, focusing on guarantees that hold uniformly at every time step after a finite burn-in period. This leads to a notion of dynamic learnability which captures how the structure of a system, such as stability, mixing, observability, and spectral properties, governs the number of observations required before reliable prediction becomes possible. We illustrate the framework in the case of linear dynamical systems, showing that accurate prediction can be achieved after finite observation without system identification, by leveraging improper methods based on spectral filtering. We survey the relationship between learning in dynamical systems and classical PAC, online, and universal prediction theories, and suggest directions for studying nonlinear and controlled systems.