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

arXiv: https://arxiv.org/abs/2509.11907 · HTML: https://mlsanity.com/paper/2509.11907/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 36 references that are in the map.
- First posted on arXiv 15 Sep 2025.

## Builds on (most cited first, 17 of 17)

- [On the Sample Complexity of the Linear Quadratic Regulator](https://mlsanity.com/paper/1710.01688/) (2017)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Active Learning for Nonlinear System Identification with Guarantees](https://mlsanity.com/paper/2006.10277/) (2020)
- [Explore First, Exploit Next: The True Shape of Regret in Bandit Problems](https://mlsanity.com/paper/1602.07182/) (2016)
- [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022)
- [Learning nonlinear dynamical systems from a single trajectory](https://mlsanity.com/paper/2004.14681/) (2020)
- [Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2002.08538/) (2020)
- [Active Learning for Identification of Linear Dynamical Systems](https://mlsanity.com/paper/2002.00495/) (2020)
- [Sample Complexity Lower Bounds for Linear System Identification](https://mlsanity.com/paper/1903.10343/) (2019)
- [Linear Systems can be Hard to Learn](https://mlsanity.com/paper/2104.01120/) (2021)
- [Task-Optimal Exploration in Linear Dynamical Systems](https://mlsanity.com/paper/2102.05214/) (2021)
- [Finite Sample Identification of Bilinear Dynamical Systems](https://mlsanity.com/paper/2208.13915/) (2022)
- [Active Learning for Control-Oriented Identification of Nonlinear Systems](https://mlsanity.com/paper/2404.09030/) (2024)
- [The Sample Complexity of Online Reinforcement Learning: A Multi-model Perspective](https://mlsanity.com/paper/2501.15910/) (2025)
- [Sample Complexity Bounds for Linear System Identification from a Finite Set](https://mlsanity.com/paper/2409.11141/) (2024)
- [Hidden Convexity in Active Learning: A Convexified Online Input Design for ARX Systems](https://mlsanity.com/paper/2509.03257/) (2025)

## Built on by (most cited first, 1 of 1)

- [Optimal Centered Active Excitation in Linear System Identification](https://mlsanity.com/paper/2604.05518/) (2026)

## Abstract (arXiv)

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.