Improved rates for prediction and identification of partially observed linear dynamical systems
This work provides improved non-asymptotic statistical rates for learning partially observed linear dynamical systems, which is significant for control theory and system identification researchers working with long-term memory systems.
This paper addresses the problem of identifying linear time-invariant dynamical systems from partial observations, especially those with long-term memory. The authors propose an algorithm that achieves a near-optimal learning rate of O(sqrt(d/T)) in H2 error, with only logarithmic dependence on memory length, given a single trajectory of length T with Gaussian observation noise.
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.