{"paper":{"title":"Lower Bounds for Finding Stationary Points II: First-Order Methods","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Aaron Sidford, John C. Duchi, Oliver Hinder, Yair Carmon","submitted_at":"2017-11-02T17:52:08Z","abstract_excerpt":"We establish lower bounds on the complexity of finding $\\epsilon$-stationary points of smooth, non-convex high-dimensional functions using first-order methods. We prove that deterministic first-order methods, even applied to arbitrarily smooth functions, cannot achieve convergence rates in $\\epsilon$ better than $\\epsilon^{-8/5}$, which is within $\\epsilon^{-1/15}\\log\\frac{1}{\\epsilon}$ of the best known rate for such methods. Moreover, for functions with Lipschitz first and second derivatives, we prove no deterministic first-order method can achieve convergence rates better than $\\epsilon^{-1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.00841","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":""},"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"}