# CLT-Optimal Parameter Error Bounds for Linear System Identification

arXiv: https://arxiv.org/abs/2604.21270 · HTML: https://mlsanity.com/paper/2604.21270/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 33 references that are in the map.
- First posted on arXiv 23 Apr 2026.

## Builds on (most cited first, 10 of 10)

- [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)
- [Finite Time Identification in Unstable Linear Systems](https://mlsanity.com/paper/1710.01852/) (2017)
- [Exact minimax risk for linear least squares, and the lower tail of sample covariance matrices](https://mlsanity.com/paper/1912.10754/) (2019)
- [Finite-time Identification of Stable Linear Systems: Optimality of the Least-Squares Estimator](https://mlsanity.com/paper/2003.07937/) (2020)
- [A Tutorial on the Non-Asymptotic Theory of System Identification](https://mlsanity.com/paper/2309.03873/) (2023)
- [Learning from many trajectories](https://mlsanity.com/paper/2203.17193/) (2022)
- [Sample Complexity Lower Bounds for Linear System Identification](https://mlsanity.com/paper/1903.10343/) (2019)
- [The noise level in linear regression with dependent data](https://mlsanity.com/paper/2305.11165/) (2023)
- [Research Program: Theory of Learning in Dynamical Systems](https://mlsanity.com/paper/2512.19410/) (2025)

## Abstract (arXiv)

There has been remarkable progress over the past decade in establishing finite-sample, non-asymptotic bounds on recovering unknown system parameters from observed system behavior. Surprisingly, however, we show that the current state-of-the-art bounds do not accurately capture the statistical complexity of system identification, even in the most fundamental setting of estimating a discrete-time linear dynamical system (LDS) via ordinary least-squares regression (OLS). Specifically, we utilize asymptotic normality to identify classes of problem instances for which current bounds overstate the squared parameter error, in both spectral and Frobenius norm, by a factor of the state-dimension of the system. Informed by this discrepancy, we then sharpen the OLS parameter error bounds via a novel second-order decomposition of the parameter error, where crucially the lower-order term is a matrix-valued martingale that we show correctly captures the CLT scaling. From our analysis we obtain finite-sample bounds for both (i) stable systems and (ii) the many-trajectories setting that match the instance-specific optimal rates up to constant factors in Frobenius norm, and polylogarithmic state-dimension factors in spectral norm.