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The paper demonstrates that existing finite-sample bounds for linear system identification using ordinary least squares (OLS) regression are suboptimal, overstating the squared parameter error by a factor of the state dimension in certain problem instances. To address this, the authors introduce a novel second-order decomposition of the parameter error, identifying a matrix-valued martingale term that captures the Central Limit Theorem (CLT) scaling. This refined analysis yields finite-sample bounds for stable systems and multi-trajectory settings that achieve instance-specific optimal rates (up to constant/polylogarithmic factors) in Frobenius and spectral norms, respectively.
Existing bounds on system identification are too pessimistic, but a new martingale-based analysis unlocks near-optimal finite-sample guarantees for parameter estimation in linear dynamical systems.
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.
