Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification
This provides a sharp theoretical analysis for linear system identification, addressing a foundational problem in control and time-series analysis.
The paper proves that the ordinary least-squares estimator achieves nearly minimax optimal performance for identifying linear dynamical systems from a single trajectory, showing that more unstable systems are easier to estimate, contrary to prior mixing-time arguments.
We prove that the ordinary least-squares (OLS) estimator attains nearly minimax optimal performance for the identification of linear dynamical systems from a single observed trajectory. Our upper bound relies on a generalization of Mendelson's small-ball method to dependent data, eschewing the use of standard mixing-time arguments. Our lower bounds reveal that these upper bounds match up to logarithmic factors. In particular, we capture the correct signal-to-noise behavior of the problem, showing that more unstable linear systems are easier to estimate. This behavior is qualitatively different from arguments which rely on mixing-time calculations that suggest that unstable systems are more difficult to estimate. We generalize our technique to provide bounds for a more general class of linear response time-series.