First-order methods for nonconvex quadratic minimization

Yair Carmon, John C. Duchi

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


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.

Original languageEnglish
Pages (from-to)395-436
Number of pages42
JournalSIAM Review
Issue number2
StatePublished - 2020
Externally publishedYes


FundersFunder number
SAIL-Toyota Center for AI Research
Stanford Graduate Fellowship
National Science FoundationNSF-CAREER-1553086
Office of Naval ResearchN00014-19-2288


    • Cubic regularization
    • Global optimization
    • Gradient descent
    • Krylov subspace methods
    • Newton's method
    • Nonasymptotic convergence
    • Nonconvex quadratics
    • Trust-region methods


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