{"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/optimal-adaptation-for-early-stopping-in","title":"Optimal adaptation for early stopping in statistical inverse problems","arxiv_id":"1606.07702","date":"2016-06-24","proceeding":null,"authors":["Gilles Blanchard","Marc Hoffmann","Markus Reiß"],"abstract":"For linear inverse problems $Y=\\mathsf{A}\\mu+\\xi$, it is classical to recover the unknown signal $\\mu$ by iterative regularisation methods $(\\widehat \\mu^{(m)}, m=0,1,\\ldots)$ and halt at a data-dependent iteration $\\tau$ using some stopping rule, typically based on a discrepancy principle, so that the weak (or prediction) squared-error $\\|\\mathsf{A}(\\widehat \\mu^{(\\tau)}-\\mu)\\|^2$ is controlled. In the context of statistical estimation with stochastic noise $\\xi$, we study oracle adaptation (that is, compared to the best possible stopping iteration) in strong squared-error $E[\\|\\hat \\mu^{(\\tau)}-\\mu\\|^2]$. For a residual-based stopping rule oracle adaptation bounds are established for general spectral regularisation methods. The proofs use bias and variance transfer techniques from weak prediction error to strong $L^2$-error, as well as convexity arguments and concentration bounds for the stochastic part. Adaptive early stopping for the Landweber method is studied in further detail and illustrated numerically.","url_abs":"https://arxiv.org/abs/1606.07702v2","url_pdf":"https://arxiv.org/pdf/1606.07702v2.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"optimal-adaptation-for-early-stopping-in","repo_url":"https://github.com/EarlyStop/EarlyStopping","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"optimal-adaptation-for-early-stopping-in","repo_url":"https://github.com/be5tan/rlystop","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}