{"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/dynamic-least-squares-regression-1","title":"The Complexity of Dynamic Least-Squares Regression","arxiv_id":"2201.00228","date":"2022-01-01","proceeding":null,"authors":["Shunhua Jiang","Binghui Peng","Omri Weinstein"],"abstract":"We settle the complexity of dynamic least-squares regression (LSR), where rows and labels $(\\mathbf{A}^{(t)}, \\mathbf{b}^{(t)})$ can be adaptively inserted and/or deleted, and the goal is to efficiently maintain an $\\epsilon$-approximate solution to $\\min_{\\mathbf{x}^{(t)}} \\| \\mathbf{A}^{(t)} \\mathbf{x}^{(t)} - \\mathbf{b}^{(t)} \\|_2$ for all $t\\in [T]$. We prove sharp separations ($d^{2-o(1)}$ vs. $\\sim d$) between the amortized update time of: (i) Fully vs. Partially dynamic $0.01$-LSR; (ii) High vs. low-accuracy LSR in the partially-dynamic (insertion-only) setting. Our lower bounds follow from a gap-amplification reduction -- reminiscent of iterative refinement -- rom the exact version of the Online Matrix Vector Conjecture (OMv) [HKNS15], to constant approximate OMv over the reals, where the $i$-th online product $\\mathbf{H}\\mathbf{v}^{(i)}$ only needs to be computed to $0.1$-relative error. All previous fine-grained reductions from OMv to its approximate versions only show hardness for inverse polynomial approximation $\\epsilon = n^{-\\omega(1)}$ (additive or multiplicative) . This result is of independent interest in fine-grained complexity and for the investigation of the OMv Conjecture, which is still widely open.","url_abs":"https://arxiv.org/abs/2201.00228v2","url_pdf":"https://arxiv.org/pdf/2201.00228v2.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":"dynamic-least-squares-regression-1","repo_url":"https://github.com/pengbinghui/dynamicl2regression","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"regression-1","task_name":"regression"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2201.00228","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}