Finite-time Identification of Stable Linear Systems: Optimality of the Least-Squares Estimator
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We provide a new finite-time analysis of the estimation error of stable linear time-invariant systems under the Ordinary Least Squares (OLS) estimator. Specifically, we characterize the sufficient number of observed samples (the length of the observed trajectory) so that the OLS is (ε,δ)-PAC, i.e. yields an estimation error less than ε with probability at least 1-δ. We show that this number matches existing sample complexity lower bound [1,2] up to universal multiplicative factors (independent of (ε,δ), of the system and of the dimension). This paper hence establishes the optimality of the OLS estimator for stable systems, a result conjectured in [1]. Our analysis of the performance of the OLS estimator is simpler, sharper, and easier to interpret than existing analyses, but is restricted to stable systems. It relies on new concentration results for the covariates matrix.
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