Regret-Optimal Filtering for Prediction and Estimation
Oron Sabag, Babak Hassibi · IEEE Transactions on Signal Processing · 2022
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 theregretby 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.