Regret-Optimal Filtering

Oron Sabag, Babak Hassibi

Research output: Contribution to journalConference articlepeer-review

6 Scopus citations


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 languageAmerican English
Pages (from-to)2629-2637
Number of pages9
JournalProceedings of Machine Learning Research
StatePublished - 2021
Externally publishedYes
Event24th International Conference on Artificial Intelligence and Statistics, AISTATS 2021 - Virtual, Online, United States
Duration: 13 Apr 202115 Apr 2021

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