Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification

TL;DR

This paper demonstrates that the ordinary least-squares estimator is nearly optimal for identifying linear dynamical systems from a single trajectory, revealing that unstable systems are easier to estimate, using a novel dependent data analysis approach.

Contribution

It introduces a new analysis technique extending Mendelson's small-ball method to dependent data, providing sharp bounds for linear system identification.

Findings

OLS estimator nearly minimax optimal for system identification

Unstable systems are easier to estimate than stable ones

New bounds match lower bounds up to logarithmic factors

Abstract

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.