{"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/laser-a-new-method-for-locally-adaptive","title":"A new approach to locally adaptive polynomial regression","arxiv_id":"2412.19802","date":"2024-12-27","proceeding":null,"authors":["Sabyasachi Chatterjee","Subhajit Goswami","Soumendu Sundar Mukherjee"],"abstract":"Adaptive bandwidth selection is a fundamental challenge in nonparametric regression. This paper introduces a new bandwidth selection procedure inspired by the optimality criteria for $\\ell_0$-penalized regression. Although similar in spirit to Lepski's method and its variants in selecting the largest interval satisfying an admissibility criterion, our approach stems from a distinct philosophy, utilizing criteria based on $\\ell_2$-norms of interval projections rather than explicit point and variance estimates. We obtain non-asymptotic risk bounds for the local polynomial regression methods based on our bandwidth selection procedure which adapt (near-)optimally to the local H\\\"{o}lder exponent of the underlying regression function simultaneously at all points in its domain. Furthermore, we show that there is a single ideal choice of a global tuning parameter in each case under which the above-mentioned local adaptivity holds. The optimal risks of our methods derive from the properties of solutions to a new ``bandwidth selection equation'' which is of independent interest. We believe that the principles underlying our approach provide a new perspective to the classical yet ever relevant problem of locally adaptive nonparametric regression.","url_abs":"https://arxiv.org/abs/2412.19802v2","url_pdf":"https://arxiv.org/pdf/2412.19802v2.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":"laser-a-new-method-for-locally-adaptive","repo_url":"https://gitlab.com/soumendu041/laser","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"philosophy","task_name":"Philosophy"},{"task_slug":"regression-1","task_name":"regression"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}