{"paper":{"title":"A Nearly-Optimal Bound for Fast Regression with $\\ell_\\infty$ Guarantee","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["cs.LG","stat.ML"],"primary_cat":"cs.DS","authors_text":"Junze Yin, Lichen Zhang, Mingquan Ye, Zhao Song","submitted_at":"2023-02-01T05:22:40Z","abstract_excerpt":"Given a matrix $A\\in \\mathbb{R}^{n\\times d}$ and a vector $b\\in \\mathbb{R}^n$, we consider the regression problem with $\\ell_\\infty$ guarantees: finding a vector $x'\\in \\mathbb{R}^d$ such that $ \\|x'-x^*\\|_\\infty \\leq \\frac{\\epsilon}{\\sqrt{d}}\\cdot \\|Ax^*-b\\|_2\\cdot \\|A^\\dagger\\|$ where $x^*=\\arg\\min_{x\\in \\mathbb{R}^d}\\|Ax-b\\|_2$. One popular approach for solving such $\\ell_2$ regression problem is via sketching: picking a structured random matrix $S\\in \\mathbb{R}^{m\\times n}$ with $m\\ll n$ and $SA$ can be quickly computed, solve the ``sketched'' regression problem $\\arg\\min_{x\\in \\mathbb{R}^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.00248","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2302.00248/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}