OnLine Learning of Linear Dynamical Systems: Exponential Forgetting in Kalman Filters
Abstract
Kalman filter is a key tool for timeseries forecasting and analysis. We show that the dependence of a prediction of Kalman filter on the past is decaying exponentially, whenever the process noise is nondegenerate. Therefore, Kalman filter may be approximated by regression on a few recent observations. Surprisingly, we also show that having some process noise is essential for the exponential decay. With no process noise, it may happen that the forecast depends on all of the past uniformly, which makes forecasting more difficult. Based on this insight, we devise an online algorithm for improper learning of a linear dynamical system (LDS), which considers only a few most recent observations. We use our decay results to provide the first regret bounds w.r.t. to Kalman filters within learning an LDS. That is, we compare the results of our algorithm to the best, in hindsight, Kalman filter for a given signal. Also, the algorithm is practical: its perupdate runtime is linear in the regression depth.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 arXiv:
 arXiv:1809.05870
 Bibcode:
 2018arXiv180905870K
 Keywords:

 Mathematics  Statistics Theory;
 Computer Science  Artificial Intelligence;
 Computer Science  Machine Learning;
 Mathematics  Optimization and Control
 EPrint:
 Proceedings of the ThirtyThird AAAI Conference on Artificial Intelligence, 2019. Pages: 40984105