CLT-Optimal Parameter Error Bounds for Linear System Identification

Yichen Zhou and Stephen Tu's paper, 'CLT-Optimal Parameter Error Bounds for Linear System Identification,' exposes a critical flaw in state-of-the-art finite-sample bounds for estimating discrete-time linear dynamical systems (LDS) via ordinary least-squares (OLS) regression. Current bounds overstate squared parameter error by a factor of the state dimension in both spectral and Frobenius norms, failing to capture true statistical complexity. The authors leverage asymptotic normality to identify this gap, then correct it with a novel second-order decomposition where the lower-order term is a matrix-valued martingale that captures CLT scaling.

This analysis yields finite-sample bounds for stable systems and many-trajectories settings that match instance-specific optimal rates up to constant factors in Frobenius norm, with only polylogarithmic state-dimension factors in spectral norm. The work bridges asymptotic and non-asymptotic theory, offering tighter guarantees for system identification in control, robotics, and time-series analysis. For practitioners, this means more reliable uncertainty quantification when learning system dynamics from limited data.