Oracle complexity of second-order methods for smooth convex optimization

Research output: Contribution to journalArticlepeer-review


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
Early online date28 May 2019
StatePublished - 1 Nov 2019


  • Oracle complexity
  • Smooth convex optimization

All Science Journal Classification (ASJC) codes

  • Software
  • General Mathematics


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