Regret-Optimal Filtering for Prediction and Estimation

We study the filtering problem of causally estimating a desired signal from a related observation signal, through the lens of regret optimization. Classical filter designs, such as
$\mathcal {H}_{2}$
(i.e., Kalman) and
$\mathcal {H}_\infty$
, minimize the average and worst-case estimation errors, respectively. As a result
$\mathcal {H}_{2}$
filters are sensitive to inaccuracies in the underlying statistical model, and
$\mathcal {H}_\infty$
filters are overly conservative since they safeguard against the worst-case scenario. In order to design filters that perform well in different noise regimes, we propose instead to minimize the
regret
by comparing the performance of the designed filter with that of a clairvoyant filter. More explicitly, we minimize the largest deviation of the squared estimation error of a causal filter from that of a non-causal filter that also has access to future observations. For the important case of signals that can be described with a linear state-space, we provide an explicit solution for the regret optimal filter in the estimation (causal) and the prediction (strictly-causal) regimes. These solutions are obtained by reducing the regret filtering problem to a Nehari problem, i.e., approximating a non-causal operator by a causal one in spectral norm. The regret-optimal filters bear some resemblance to Kalman and
$\mathcal {H}_\infty$
filters: they are expressed as state-space models, inherit the finite dimension of the original state-space, and their solutions require solving algebraic Riccati equations. Numerical simulations demonstrate that regret minimization inherently interpolates between the performances of the
$\mathcal {H}_{2}$
and
$\mathcal {H}_\infty$
filters and is thus a viable approach for filter design.