CLT-Optimal Parameter Error Bounds for Linear System Identification

2026-04-23

AI Summary

Existing finite-sample error bounds in linear system identification are significantly loose due to overestimation of the influence of state dimensionality. This work reveals, via the central limit theorem, that ordinary least squares (OLS) estimation incurs error bounds whose looseness scales proportionally with the state dimension. To address this, the paper introduces a novel second-order error decomposition that leverages a matrix-valued martingale structure to characterize lower-order terms. As a result, it establishes non-asymptotic error bounds that are constant-factor tight in the Frobenius norm and depend on the state dimension only through a polylogarithmic factor in the spectral norm. For stable systems and multi-trajectory settings, these bounds approach instance-optimal rates, markedly improving upon the tightness of existing results.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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.

Problem

• linear system identification
• parameter error bounds
• ordinary least-squares
• statistical complexity
• non-asymptotic bounds

Innovation

• linear system identification
• non-asymptotic bounds
• ordinary least squares
• matrix-valued martingale
• CLT scaling