{"paper":{"title":"On the Randomized Complexity of Minimizing a Convex Quadratic Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.OC","stat.ML"],"primary_cat":"cs.LG","authors_text":"Max Simchowitz","submitted_at":"2018-07-24T23:23:49Z","abstract_excerpt":"Minimizing a convex, quadratic objective of the form $f_{\\mathbf{A},\\mathbf{b}}(x) := \\frac{1}{2}x^\\top \\mathbf{A} x - \\langle \\mathbf{b}, x \\rangle$ for $\\mathbf{A} \\succ 0 $ is a fundamental problem in machine learning and optimization. In this work, we prove gradient-query complexity lower bounds for minimizing convex quadratic functions which apply to both deterministic and \\emph{randomized} algorithms. Specifically, for $\\kappa > 1$, we exhibit a distribution over $(\\mathbf{A},\\mathbf{b})$ with condition number $\\mathrm{cond}(\\mathbf{A}) \\le \\kappa$, such that any \\emph{randomized} algori"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.09386","kind":"arxiv","version":7},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}