Online Learning of the Kalman Filter with Logarithmic Regret

TL;DR

This paper introduces an online learning algorithm for the Kalman filter that guarantees logarithmic regret in prediction accuracy, even for non-explosive systems, advancing the theoretical understanding of adaptive filtering.

Contribution

It provides the first logarithmic regret guarantees for the Kalman filter in an online setting, applicable to non-explosive and marginally stable systems.

Findings

Achieves polylogarithmic regret with high probability.

Applicable to systems with unknown noise statistics.

Extends analysis to non-explosive, marginally stable systems.

Abstract

In this paper, we consider the problem of predicting observations generated online by an unknown, partially observed linear system, which is driven by stochastic noise. For such systems the optimal predictor in the mean square sense is the celebrated Kalman filter, which can be explicitly computed when the system model is known. When the system model is unknown, we have to learn how to predict observations online based on finite data, suffering possibly a non-zero regret with respect to the Kalman filter's prediction. We show that it is possible to achieve a regret of the order of $\mathrm{poly}\log(N)$ with high probability, where $N$ is the number of observations collected. Our work is the first to provide logarithmic regret guarantees for the widely used Kalman filter. This is achieved using an online least-squares algorithm, which exploits the approximately linear relation between future observations and past observations. The regret analysis is based on the stability properties of the Kalman filter, recent statistical tools for finite sample analysis of system identification, and classical results for the analysis of least-squares algorithms for time series. Our regret analysis can also be applied for state prediction of the hidden state, in the case of unknown noise statistics but known state-space basis. A fundamental technical contribution is that our bounds hold even for the class of non-explosive systems, which includes the class of marginally stable systems, which was an open problem for the case of online prediction under stochastic noise.