On-Line Learning of Linear Dynamical Systems: Exponential Forgetting in Kalman Filters

TL;DR

This paper demonstrates that Kalman filter predictions depend exponentially less on distant past observations when process noise is present, enabling efficient online learning of linear dynamical systems with regret guarantees.

Contribution

It proves exponential decay of Kalman filter dependence on past when process noise exists and introduces a practical online learning algorithm with regret bounds for LDS.

Findings

Kalman filter dependence on past decays exponentially with process noise.

Without process noise, dependence on the entire past can be uniform.

Proposed algorithm is practical with linear per-update runtime.

Abstract

Kalman filter is a key tool for time-series 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 non-degenerate. 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 on-line 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 per-update run-time is linear in the regression depth.