## Abstract

We consider the problem of filtering in linear state-space models (e.g., the Kalman filter setting) through the lens of regret optimization. Specifically, we study the problem of causally estimating a desired signal, generated by a linear state-space model driven by process noise, based on noisy observations of a related observation process. We define a novel regret criterion for estimator design as the difference of the estimation error energies between a clairvoyant estimator that has access to all future observations (a so-called smoother) and a causal one that only has access to current and past observations. The regret-optimal estimator is the causal estimator that minimizes the worst-case regret across all bounded-energy noise sequences. We provide a solution for the regret filtering problem at two levels. First, an horizon-independent solution at the operator level is obtained by reducing the regret to the well-known Nehari problem. Secondly, our main result for state-space models is an explicit estimator that achieves the optimal regret. The regret-optimal estimator is represented as a finite-dimensional state-space whose parameters can be computed by solving three Riccati equations and a single Lyapunov equation. We demonstrate the applicability and efficacy of the estimator in a variety of problems and observe that the estimator has average and worst-case performances that are simultaneously close to their optimal values.

Original language | American English |
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Pages (from-to) | 2629-2637 |

Number of pages | 9 |

Journal | Proceedings of Machine Learning Research |

Volume | 130 |

State | Published - 2021 |

Externally published | Yes |

Event | 24th International Conference on Artificial Intelligence and Statistics, AISTATS 2021 - Virtual, Online, United States Duration: 13 Apr 2021 → 15 Apr 2021 |

### Bibliographical note

Publisher Copyright:Copyright © 2021 by the author(s)