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  • 标题:Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions
  • 本地全文:下载
  • 作者:Ioannis Panageas ; Georgios Piliouras
  • 期刊名称:LIPIcs : Leibniz International Proceedings in Informatics
  • 电子版ISSN:1868-8969
  • 出版年度:2017
  • 卷号:67
  • 页码:2:1-2:12
  • DOI:10.4230/LIPIcs.ITCS.2017.2
  • 出版社:Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
  • 摘要:Given a twice continuously differentiable cost function f, we prove that the set of initial conditions so that gradient descent converges to saddle points where \nabla^2 f has at least one strictly negative eigenvalue, has (Lebesgue) measure zero, even for cost functions f with non-isolated critical points, answering an open question in [Lee, Simchowitz, Jordan, Recht, COLT 2016]. Moreover, this result extends to forward-invariant convex subspaces, allowing for weak (non-globally Lipschitz) smoothness assumptions. Finally, we produce an upper bound on the allowable step-size.
  • 关键词:Gradient Descent; Center-stable manifold; Saddle points; Hessian
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