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Research Program: Theory of Learning in Dynamical Systems

Elad Hazan, Shai Shalev Shwartz, Nathan Srebro

TL;DR

The paper advocates a finite-sample theory of learnability for dynamical systems by recasting prediction in a next-token framework and emphasizing structural dynamics (stability, mixing, observability, spectral properties) over distributional assumptions. It introduces dynamic learnability with burn-in time $T(\varepsilon)$ that guarantees uniform, time-invariant predictive risk after the burn-in, and demonstrates improper learnability without system identification via spectral filtering in symmetric linear dynamical systems. By tying dynamical learning to stochastic-process perspectives, Bayes-optimal predictors, and realizable/agnostic settings, the work unifies classical PAC/online viewpoints while addressing latent state and memory. The paper also surveys connections to control theory, operator methods, and future directions, including nonlinear dynamics, chaotic systems, data assimilation, and training dynamics of large language models, outlining concrete research programs and open questions for algorithmic efficiency and spectral complexity.

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.

Research Program: Theory of Learning in Dynamical Systems

TL;DR

The paper advocates a finite-sample theory of learnability for dynamical systems by recasting prediction in a next-token framework and emphasizing structural dynamics (stability, mixing, observability, spectral properties) over distributional assumptions. It introduces dynamic learnability with burn-in time that guarantees uniform, time-invariant predictive risk after the burn-in, and demonstrates improper learnability without system identification via spectral filtering in symmetric linear dynamical systems. By tying dynamical learning to stochastic-process perspectives, Bayes-optimal predictors, and realizable/agnostic settings, the work unifies classical PAC/online viewpoints while addressing latent state and memory. The paper also surveys connections to control theory, operator methods, and future directions, including nonlinear dynamics, chaotic systems, data assimilation, and training dynamics of large language models, outlining concrete research programs and open questions for algorithmic efficiency and spectral complexity.

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.
Paper Structure (23 sections, 1 theorem, 16 equations, 1 figure)

This paper contains 23 sections, 1 theorem, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. What we mean by dynamic learnability.
  3. Departures from classical learning theory.
  4. Bridging learning theory and control.
  5. Why is next-token prediction a good metric?
  6. Improper Dynamic Learnability without Identification
  7. Spectral Filtering for Symmetric Linear Dynamical Systems
  8. Symmetric linear dynamical systems.
  9. Universal Spectral Basis.
  10. Spectral Complexity and Predictability
  11. Learnability in Dynamical Systems and Stochastic Processes
  12. Dynamical Systems
  13. Stochastic Processes
  14. Dynamical systems as stochastic processes.
  15. Examples of model classes.
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let the system eq:symmetric-lds satisfy $\|A\|_2 \le 1$ and be driven by independent sub-Gaussian noise. Let $\phi_1,\ldots,\phi_m$ be the top $m$ eigenvectors of the Hilbert matrix $\mathsf{H}_T$. There exists a choice of filter count $m = \Theta(\log T \log(1/\varepsilon))$ such that the spectral Specifically, for all $t > T(\varepsilon)$, the expected excess risk is bounded by:

Figures (1)

  • Figure 1: Example spectral filters $\phi_1,\phi_3,\phi_5,\phi_{10},\phi_{20}$ obtained as the top eigenvectors of a Hilbert or Hankel matrix.

Theorems & Definitions (5)

  • Theorem 2.1: Dynamic Learnability of Symmetric LDS
  • proof : Proof Sketch of Theorem \ref{['thm:symmetric-sf']}
  • Definition 3.1: Dynamic Learnability of Dynamical Systems
  • Definition 3.2: Realizable (well-specified) dynamic learnability
  • Definition 3.3: Agnostic dynamic learnability