Oracle complexity of second-order methods for smooth convex optimization

Yossi Arjevani, Ohad Shamir, Ron Shiff*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

37 Scopus citations


Second-order methods, which utilize gradients as well as Hessians to optimize a given function, are of major importance in mathematical optimization. In this work, we prove tight bounds on the oracle complexity of such methods for smooth convex functions, or equivalently, the worst-case number of iterations required to optimize such functions to a given accuracy. In particular, these bounds indicate when such methods can or cannot improve on gradient-based methods, whose oracle complexity is much better understood. We also provide generalizations of our results to higher-order methods.

Original languageAmerican English
Pages (from-to)327-360
Number of pages34
JournalMathematical Programming
Issue number1-2
StatePublished - 1 Nov 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.


  • Oracle complexity
  • Smooth convex optimization


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