Finite-time Identification of Stable Linear Systems: Optimality of the Least-Squares Estimator
This establishes the optimality of OLS for stable linear systems, resolving a conjecture and providing a simpler analysis for researchers in control theory and system identification.
The paper tackles the problem of finite-time identification of stable linear systems, showing that the Ordinary Least Squares (OLS) estimator achieves optimal sample complexity, matching existing lower bounds up to universal multiplicative factors.
We present a new finite-time analysis of the estimation error of the Ordinary Least Squares (OLS) estimator for stable linear time-invariant systems. We characterize the number of observed samples (the length of the observed trajectory) sufficient for the OLS estimator to be $(\varepsilon,δ)$-PAC, i.e., to yield an estimation error less than $\varepsilon$ with probability at least $1-δ$. We show that this number matches existing sample complexity lower bounds [1,2] up to universal multiplicative factors (independent of ($\varepsilon,δ)$ and of the system). 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. It relies on new concentration results for the covariates matrix.