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

2025-09-15

AI Summary

This work addresses system identification of linear dynamical systems under a finite hypothesis class, with emphasis on how excitation properties of the input signal affect sample complexity. We introduce a novel, hypothesis-class-adapted persistent excitation condition that relaxes conventional global excitation requirements. Building upon this, we design an actively learning algorithm with rigorous theoretical guarantees and derive tight upper and lower bounds on sample complexity—both exhibiting identical dependence on key problem parameters, thereby characterizing the fundamental performance limit for this setting for the first time. Our approach integrates statistical learning theory and system identification, employing a modular analysis framework comprising lower-bound derivation, excitation-condition construction, policy optimization, and probabilistic consistency proof. Extensive simulations demonstrate the algorithm’s superior identification accuracy and sampling efficiency compared to existing methods.

Technology Category

• Machine Learning: Active Learning
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 acs{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.

Problem

• Sample complexity limits for identifying linear dynamical systems
• Effect of excitation input on system identification accuracy
• Active learning algorithm with optimal sequential excitation design

Innovation

• Sample complexity bounds for system identification
• Persistent excitation condition for finite hypothesis
• Active learning algorithm with optimal excitation