@article{2b972523737a4eaa9a49f0e1b2787ad3,
title = "First-order methods for nonconvex quadratic minimization",
abstract = "We consider minimization of indefinite quadratics with either trust-region (norm) constraints or cubic regularization. Despite the nonconvexity of these problems we prove that, under mild assumptions, gradient descent converges to their global solutions and give a nonasymptotic rate of convergence for the cubic variant. We also consider Krylov subspace solutions and establish sharp convergence guarantees to the solutions of both trust-region and cubic-regularized problems. Our rates mirror the behavior of these methods on convex quadratics and eigenvector problems, highlighting their scalability. When we use Krylov subspace solutions to approximate the cubic-regularized Newton step, our results recover the strongest known convergence guarantees to approximate second-order stationary points of general smooth nonconvex functions.",
keywords = "Cubic regularization, Global optimization, Gradient descent, Krylov subspace methods, Newton's method, Nonasymptotic convergence, Nonconvex quadratics, Trust-region methods",
author = "Yair Carmon and Duchi, {John C.}",
note = "Publisher Copyright: {\textcopyright} 2020 Society for Industrial and Applied Mathematics Publications. All rights reserved.",
year = "2020",
doi = "10.1137/20M1321759",
language = "אנגלית",
volume = "62",
pages = "395--436",
journal = "SIAM Review",
issn = "0036-1445",
publisher = "Society for Industrial and Applied Mathematics (SIAM)",
number = "2",
}