Accelerated methods for nonconvex optimization

Yair Carmon, John C. Duchi, Oliver Hinder, Aaron Sidford

Research output: Contribution to journalArticlepeer-review

122 Scopus citations


We present an accelerated gradient method for nonconvex optimization problems with Lipschitz continuous first and second derivatives. In a time O(7/4 log(1/)), the method finds an -stationary point, meaning a point x such that ∇f(x) ≤ . The method improves upon the O( 2) complexity of gradient descent and provides the additional second-order guarantee that λmin(∇2f(x)) −1/2 for the computed x. Furthermore, our method is Hessian free, i.e., it only requires gradient computations, and is therefore suitable for large-scale applications.

Original languageEnglish
Pages (from-to)1751-1772
Number of pages22
JournalSIAM Journal on Optimization
Issue number2
StatePublished - 2018
Externally publishedYes


FundersFunder number
Stanford Graduate Fellowship
National Science FoundationNSF-CAREER-1553086
National Science Foundation1553086


    • Accelerated gradient descent
    • Convergence rate
    • Lanczos method
    • Negative curvature
    • Nonlinear optimization
    • Second-order stationarity
    • Semiconvexity


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