  We consider the problem of filtering in linear state-space models (e.g., the
Kalman filter setting) through the lens of regret optimization. Different
assumptions on the driving disturbance and the observation noise sequences give
rise to different estimators: in the stochastic setting to the celebrated
Kalman filter, and in the deterministic setting of bounded energy disturbances
to $H_\infty$ estimators. In this work, we formulate a novel criterion for
filter design based on the concept of regret between the estimation error
energy of 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 chosen to minimize this
worst-case difference across all bounded-energy noise sequences. The resulting
estimator is adaptive in the sense that it aims to mimic the behavior of the
clairvoyant estimator, irrespective of what the realization of the noise will
be and thus interpolates between the stochastic and deterministic approaches.
We provide a solution for the regret estimation problem at two different
levels. First, we provide a solution at the operator level by reducing it to
the Nehari problem. Second, for state-space models, we explicitly find the
estimator that achieves the optimal regret. From a computational perspective,
the regret-optimal estimator can be easily implemented by solving three Riccati
equations and a single Lyapunov equation. For a state-space model of dimension
$n$, the regret-optimal estimator has a state-space structure of dimension
$3n$. 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. We
therefore argue that regret-optimality is a viable approach to estimator
design.
