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A finite-sample generalization bound for stable LPV systems

Daniel Racz, Martin Gonzalez, Mihaly Petreczky, Andras Benczur, Balint Daroczy

TL;DR

This work derives a time-horizon–independent PAC generalization bound for stable continuous-time LPV-SSA models by leveraging a Volterra-series representation and a $\lambda$-weighted $H_2$ norm to bound the Rademacher complexity of the hypothesis class. The bound scales as $O(c/\sqrt{N})$ with $c = 2K_{\ell}\max\{L_u(n_p+1)c_E, c_y\}$ and $R(\delta)=c\left(2+4\sqrt{2\log(4/\delta)}\right)$, and holds uniformly over all models in the class with probability at least $1-\delta$, independent of the integration horizon $T$. The approach provides practical data-driven guarantees for unseen trajectories and yields a minimum sample size $N_m=(R(\delta)/\epsilon)^2$ to achieve a target accuracy, with extensions to multi-output scenarios. By combining Volterra-series representations with Rademacher-complexity bounds, the paper contributes a theoretically principled, applicable framework for PAC-style generalization in LPV-based system identification and related dynamical modeling tasks.

Abstract

One of the main theoretical challenges in learning dynamical systems from data is providing upper bounds on the generalization error, that is, the difference between the expected prediction error and the empirical prediction error measured on some finite sample. In machine learning, a popular class of such bounds are the so-called Probably Approximately Correct (PAC) bounds. In this paper, we derive a PAC bound for stable continuous-time linear parameter-varying (LPV) systems. Our bound depends on the H2 norm of the chosen class of the LPV systems, but does not depend on the time interval for which the signals are considered.

A finite-sample generalization bound for stable LPV systems

TL;DR

This work derives a time-horizon–independent PAC generalization bound for stable continuous-time LPV-SSA models by leveraging a Volterra-series representation and a -weighted norm to bound the Rademacher complexity of the hypothesis class. The bound scales as with and , and holds uniformly over all models in the class with probability at least , independent of the integration horizon . The approach provides practical data-driven guarantees for unseen trajectories and yields a minimum sample size to achieve a target accuracy, with extensions to multi-output scenarios. By combining Volterra-series representations with Rademacher-complexity bounds, the paper contributes a theoretically principled, applicable framework for PAC-style generalization in LPV-based system identification and related dynamical modeling tasks.

Abstract

One of the main theoretical challenges in learning dynamical systems from data is providing upper bounds on the generalization error, that is, the difference between the expected prediction error and the empirical prediction error measured on some finite sample. In machine learning, a popular class of such bounds are the so-called Probably Approximately Correct (PAC) bounds. In this paper, we derive a PAC bound for stable continuous-time linear parameter-varying (LPV) systems. Our bound depends on the H2 norm of the chosen class of the LPV systems, but does not depend on the time interval for which the signals are considered.
Paper Structure (13 sections, 7 theorems, 33 equations, 1 figure)

This paper contains 13 sections, 7 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. LPV systems
  3. Learning problem and generalization
  4. Technical preliminaries and assumptions
  5. Main result
  6. Proof of Theorem \ref{['thm:main']}
  7. Experiments
  8. Conclusion
  9. Appendix A: Proof of Lemma \ref{['lemma:volterra']}
  10. Appendix B: proof of Lemma \ref{['lemma:h2norm']} and bounding the $H_2$ norms
  11. Appendix C: multi output
  12. Appendix D: Computing $\theta$ in numerical example
  13. Appendix E: difference between minimal prediction error models and best possible models

Key Result

Lemma IV.1

For every $(u,p) \in \mathcal{U} \times \mathcal{P}$, $\lambda \ge 0$, the output of $\Sigma$ at time $T$ admits the following representation:

Figures (1)

  • Figure 1: Difference between the estimated and the true error. The estimated error is the sum of the empirical error measured on a finite set with $N$ elements and the PAC bound.

Theorems & Definitions (9)

  • Remark III.1
  • Lemma IV.1
  • Remark IV.2
  • Lemma IV.4
  • Theorem V.1: Main result
  • Theorem VI.1
  • Lemma VI.2
  • Lemma X.2
  • Corollary XIII.1