{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/regret-optimal-filtering","title":"Regret-Optimal Filtering for Prediction and Estimation","arxiv_id":"2101.10357","date":"2021-01-25","proceeding":null,"authors":["Oron Sabag","Babak Hassibi"],"abstract":"The filtering problem of causally estimating a desired signal from a related observation signal is investigated through the lens of regret optimization. Classical filter designs, such as $\\mathcal H_2$ (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. We propose instead to minimize the \\emph{regret} in order to design filters that perform well in different noise regimes by comparing their performance 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 has access to future observations. In this sense, the regret-optimal filter will have the best competitive performance with respect to the non-causal benchmark filter no matter what the true signal and the observation process are. For the important case of signals that can be described with a time-invariant state-space, we provide an explicit construction 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 $H_\\infty$ 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 $H_2$ and $H_\\infty$ filters and is thus a viable approach for filter design.","url_abs":"https://arxiv.org/abs/2101.10357v3","url_pdf":"https://arxiv.org/pdf/2101.10357v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"regret-optimal-filtering","repo_url":"https://github.com/oronsabag/regret-optimal-filtering","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"unanswered"}}],"tasks":[{"task_slug":"prediction","task_name":"Prediction"},{"task_slug":"state-space-models","task_name":"State Space Models"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2101.10357","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}