Improved rates for prediction and identification of partially observed linear dynamical systems

TL;DR

This paper introduces a near-optimal algorithm for identifying linear dynamical systems from partial observations, achieving fast learning rates that depend on system order rather than memory length, with robust theoretical guarantees.

Contribution

The authors develop a novel multi-scale low-rank approximation algorithm with sharp concentration bounds, improving learning rates for partially observed linear systems.

Findings

Achieves $ ilde{O}( oot{d}{T})$ error rate in $ ext{H}_2$ norm

Logarithmic dependence on memory length in sample complexity

Provides sharper concentration bounds for correlated inputs

Abstract

Identification of a linear time-invariant dynamical system from partial observations is a fundamental problem in control theory. Particularly challenging are systems exhibiting long-term memory. A natural question is how learn such systems with non-asymptotic statistical rates depending on the inherent dimensionality (order) $d$ of the system, rather than on the possibly much larger memory length. We propose an algorithm that given a single trajectory of length $T$ with gaussian observation noise, learns the system with a near-optimal rate of $\widetilde O\left(\sqrt\frac{d}{T}\right)$ in $\mathcal{H}_2$ error, with only logarithmic, rather than polynomial dependence on memory length. We also give bounds under process noise and improved bounds for learning a realization of the system. Our algorithm is based on multi-scale low-rank approximation: SVD applied to Hankel matrices of geometrically increasing sizes. Our analysis relies on careful application of concentration bounds on the Fourier domain -- we give sharper concentration bounds for sample covariance of correlated inputs and for $\mathcal H_\infty$ norm estimation, which may be of independent interest.