CLT-Optimal Parameter Error Bounds for Linear System Identification

TL;DR

This paper refines finite-sample bounds for linear system identification, revealing that previous bounds overestimate errors and providing sharper, instance-specific error bounds that match optimal rates.

Contribution

It introduces a novel second-order decomposition of parameter error, improving bounds for system identification by accurately capturing the CLT scaling.

Findings

Current bounds overstate error by a factor of the system's state-dimension.

New bounds match optimal rates up to constants and polylog factors.

Analysis applies to both stable systems and multiple-trajectory settings.

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.