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High Effort, Low Gain: Fundamental Limits of Active Learning for Linear Dynamical Systems

Nicolas Chatzikiriakos, Kevin Jamieson, Andrea Iannelli

TL;DR

This work analyzes the identification of an unknown discrete-time LTI system from a finite hypothesis class using a single correlated trajectory. It derives instance-dependent lower bounds showing the potential benefit of oracle experiment design and introduces a modular Persistency of Excitation (PE) framework to obtain finite-sample upper bounds for any excitation satisfying PE. A CE-based sequential input design algorithm is proposed to generate informative excitations by leveraging the current estimate, with rigorous PE-based guarantees that the true system is identified with high probability. Numerical experiments demonstrate that active input design can yield substantial speedups in structured problem settings and that naive isotropic excitations can be significantly suboptimal, highlighting the practical value of the proposed framework.

Abstract

In this work, we consider the problem of identifying an unknown linear dynamical system given a finite hypothesis class. In particular, we analyze the effect of the excitation input on the sample complexity of identifying the true system with high probability. To this end, we present sample complexity lower bounds that capture the choice of the selected excitation input. The sample complexity lower bound gives rise to a system theoretic condition to determine the potential benefit of experiment design. Informed by the analysis of the sample complexity lower bound, we propose a persistent excitation (PE) condition tailored to the considered setting, which we then use to establish sample complexity upper bounds. Notably, the PE condition is weaker than in the case of an infinite hypothesis class and allows analyzing different excitation inputs modularly. Crucially, the lower and upper bounds share the same dependency on key problem parameters. Finally, we leverage these insights to propose an active learning algorithm that sequentially excites the system optimally with respect to the current estimate, and provide sample complexity guarantees for the presented algorithm. Concluding simulations showcase the effectiveness of the proposed algorithm.

High Effort, Low Gain: Fundamental Limits of Active Learning for Linear Dynamical Systems

TL;DR

This work analyzes the identification of an unknown discrete-time LTI system from a finite hypothesis class using a single correlated trajectory. It derives instance-dependent lower bounds showing the potential benefit of oracle experiment design and introduces a modular Persistency of Excitation (PE) framework to obtain finite-sample upper bounds for any excitation satisfying PE. A CE-based sequential input design algorithm is proposed to generate informative excitations by leveraging the current estimate, with rigorous PE-based guarantees that the true system is identified with high probability. Numerical experiments demonstrate that active input design can yield substantial speedups in structured problem settings and that naive isotropic excitations can be significantly suboptimal, highlighting the practical value of the proposed framework.

Abstract

In this work, we consider the problem of identifying an unknown linear dynamical system given a finite hypothesis class. In particular, we analyze the effect of the excitation input on the sample complexity of identifying the true system with high probability. To this end, we present sample complexity lower bounds that capture the choice of the selected excitation input. The sample complexity lower bound gives rise to a system theoretic condition to determine the potential benefit of experiment design. Informed by the analysis of the sample complexity lower bound, we propose a persistent excitation (PE) condition tailored to the considered setting, which we then use to establish sample complexity upper bounds. Notably, the PE condition is weaker than in the case of an infinite hypothesis class and allows analyzing different excitation inputs modularly. Crucially, the lower and upper bounds share the same dependency on key problem parameters. Finally, we leverage these insights to propose an active learning algorithm that sequentially excites the system optimally with respect to the current estimate, and provide sample complexity guarantees for the presented algorithm. Concluding simulations showcase the effectiveness of the proposed algorithm.

Paper Structure

This paper contains 28 sections, 22 theorems, 130 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Can active learning algorithms enhance identification?
  4. Sample complexity lower bounds
  5. Sample complexity upper bounds
  6. Persistency of Excitation
  7. High probability identification with finite samples
  8. Sequential input design algorithm
  9. Numerical experiments
  10. Conclusion
  11. Related Works
  12. Proofs for sample complexity lower bounds
  13. Proof of Theorem \ref{['th:SampleComplexLowerGeneral']}
  14. Proof of Corollary \ref{['co:lowerBoundGaussian']}
  15. Existence of the optimal solution to \ref{['eq:defOptU']}
  16. ...and 13 more sections

Key Result

Theorem 3.1

Consider the unknown dynamical system eq:TrueSysEvo with $x(0) = 0$ and the set $\mathcal{S}$ defined in eq:defSet. Then for any (potentially random) excitation input sequenceRandom and deterministic input sequences are denoted identically. The interpretation will be clear from the context.$U\in \ma Furthermore, under Assumption ass:inputConstr, the lower bound is minimized by the excitation input

Figures (2)

  • Figure 1: Mean and $\frac{\sigma}{2}$-band of the likelihood of $\theta_*$ given the data for different excitation strategies and settings.
  • Figure 2: Mean and $\sigma$-band of the likelihood of $\theta_*$ given the data for different data collection strategies for the motivating example.

Theorems & Definitions (42)

  • Definition 1: $\delta$-correct algorithms
  • Theorem 3.1
  • Remark 3.1
  • Corollary 3.2
  • Example 3.1
  • Definition 2: PE
  • Lemma 3.3: PE for isotropic Gaussian inputs
  • Lemma 3.4: PE for optimal oracle excitation
  • Remark 3.2
  • Theorem 3.5
  • ...and 32 more