Second-Order Information in Non-Convex Stochastic Optimization: Power and Limitations

Yossi Arjevani*, Yair Carmon, John C. Duchi, Dylan J. Foster, Ayush Sekhari, Karthik Sridharan

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

27 Scopus citations


We design an algorithm which finds an ε-approximate stationary point (with k∇F(x)k ≤ ε) using O(ε3) stochastic gradient and Hessian-vector products, matching guarantees that were previously available only under a stronger assumption of access to multiple queries with the same random seed. We prove a lower bound which establishes that this rate is optimal and—surprisingly—that it cannot be improved using stochastic pth order methods for any p ≥ 2, even when the first p derivatives of the objective are Lipschitz. Together, these results characterize the complexity of non-convex stochastic optimization with second-order methods and beyond. Expanding our scope to the oracle complexity of finding (ε, γ)-approximate second-order stationary points, we establish nearly matching upper and lower bounds for stochastic second-order methods. Our lower bounds here are novel even in the noiseless case.

Original languageAmerican English
Pages (from-to)242-299
Number of pages58
JournalProceedings of Machine Learning Research
StatePublished - 2020
Externally publishedYes
Event33rd Conference on Learning Theory, COLT 2020 - Virtual, Online, Austria
Duration: 9 Jul 202012 Jul 2020

Bibliographical note

Funding Information:
We thank Blake Woodworth and Nati Srebo for helpful discussions. YA acknowledges partial support from the Sloan Foundation and Samsung Research. JCD acknowledges support from the NSF CAREER award CCF-1553086, ONR YIP N00014-19-2288, Sloan Foundation, NSF HDR 1934578 (Stanford Data Science Collaboratory), and the DAWN Consortium. DF acknowledges the support of TRIPODS award 1740751. KS acknowledges support from NSF CAREER Award 1750575 and a Sloan Research Fellowship.

Publisher Copyright:
© 2020 Y. Arjevani, Y. Carmon, J. C. Duchi, D. J. Foster, A. Sekhari & K. Sridharan.


  • Hessian-vector products
  • Stochastic optimization
  • non-convex optimization
  • second-order methods
  • variance reduction


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