{"paper":{"title":"Third-order Smoothness Helps: Even Faster Stochastic Optimization Algorithms for Finding Local Minima","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"math.OC","authors_text":"Pan Xu, Quanquan Gu, Yaodong Yu","submitted_at":"2017-12-18T18:57:44Z","abstract_excerpt":"We propose stochastic optimization algorithms that can find local minima faster than existing algorithms for nonconvex optimization problems, by exploiting the third-order smoothness to escape non-degenerate saddle points more efficiently. More specifically, the proposed algorithm only needs $\\tilde{O}(\\epsilon^{-10/3})$ stochastic gradient evaluations to converge to an approximate local minimum $\\mathbf{x}$, which satisfies $\\|\\nabla f(\\mathbf{x})\\|_2\\leq\\epsilon$ and $\\lambda_{\\min}(\\nabla^2 f(\\mathbf{x}))\\geq -\\sqrt{\\epsilon}$ in the general stochastic optimization setting, where $\\tilde{O}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.06585","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"}