Oracle complexity of second-order methods for smooth convex optimization

  • Yossi Arjevani
  • Ohad Shamir
  • Ron ShiffEmail author
Full Length Paper Series A


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.


Smooth convex optimization Oracle complexity 

Mathematics Subject Classification

90C25 65K05 49M37 



We thank Yurii Nesterov for several helpful comments on a preliminary version of this paper, as well as Naman Agarwal, Elad Hazan and Zeyuan Allen-Zhu for informing us about the A-NPE algorithm of [10].


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceWeizmann Institute of ScienceRehovotIsrael

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