{"paper":{"title":"Lower Bounds for Finding Stationary Points I","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":"2017-10-31T17:37:07Z","abstract_excerpt":"We prove lower bounds on the complexity of finding $\\epsilon$-stationary points (points $x$ such that $\\|\\nabla f(x)\\| \\le \\epsilon$) of smooth, high-dimensional, and potentially non-convex functions $f$. We consider oracle-based complexity measures, where an algorithm is given access to the value and all derivatives of $f$ at a query point $x$. We show that for any (potentially randomized) algorithm $\\mathsf{A}$, there exists a function $f$ with Lipschitz $p$th order derivatives such that $\\mathsf{A}$ requires at least $\\epsilon^{-(p+1)/p}$ queries to find an $\\epsilon$-stationary point. Our "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.11606","kind":"arxiv","version":3},"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/1710.11606/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"}