Skip to main content

One-Point Gradient-Free Methods for Smooth and Non-smooth Saddle-Point Problems

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 12755)


In this paper, we analyze gradient-free methods with one-point feedback for stochastic saddle point problems \(\min _{x}\max _{y} \varphi (x, y)\). For non-smooth and smooth cases, we present an analysis in a general geometric setup with the arbitrary Bregman divergence. For problems with higher order smoothness, the analysis is carried out only in the Euclidean case. The estimates we have obtained repeat the best currently known estimates of gradient-free methods with one-point feedback for problems of imagining a convex or strongly convex function. The paper uses three main approaches to recovering the gradient through finite differences: standard with a random direction, as well as its modifications with kernels and residual feedback. We also provide experiments to compare these approaches for the matrix game.


  • Saddle-point problem
  • Zeroth order method
  • One-point feedback
  • Stochastic optimization

The research of A. Beznosikov and A. Gasnikov was supported by Russian Science Foundation (project No. 21-71-30005).

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions


  1. Akhavan, A., Pontil, M., Tsybakov, A.B.: Exploiting higher order smoothness in derivative-free optimization and continuous bandits. arXiv preprint arXiv:2006.07862 (2020)

  2. Bach, F., Perchet, V.: Highly-smooth zero-th order online optimization. In: Conference on Learning Theory, pp. 257–283. PMLR (2016)

    Google Scholar 

  3. Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization: analysis, algorithms, and engineering applications (2019)

    Google Scholar 

  4. Beznosikov, A., Gorbunov, E., Gasnikov, A.: Derivative-free method for composite optimization with applications to decentralized distributed optimization. arXiv preprint arXiv:1911.10645 (2019)

  5. Beznosikov, A., Novitskii, V., Gasnikov, A.: One-point gradient-free methods for smooth and non-smooth saddle-point problems. arXiv preprint arXiv:2103.00321 (2021)

  6. Beznosikov, A., Sadiev, A., Gasnikov, A.: Gradient-free methods for saddle-point problem. arXiv preprint arXiv:2005.05913 (2020)

  7. Carmon, Y., Jin, Y., Sidford, A., Tian, K.: Coordinate methods for matrix games. arXiv preprint arXiv:2009.08447 (2020)

  8. Chen, P.Y., Zhang, H., Sharma, Y., Yi, J., Hsieh, C.J.: Zoo. In: Proceedings of the 10th ACM Workshop on Artificial Intelligence and Security - AISec 2017 (2017).

  9. Duchi, J.C., Jordan, M.I., Wainwright, M.J., Wibisono, A.: Optimal rates for zero-order convex optimization: the power of two function evaluations. arXiv preprint arXiv:1312.2139 (2013)

  10. Fazel, M., Ge, R., Kakade, S., Mesbahi, M.: Global convergence of policy gradient methods for the linear quadratic regulator. In: International Conference on Machine Learning, pp. 1467–1476. PMLR (2018)

    Google Scholar 

  11. Gasnikov, A.V., Krymova, E.A., Lagunovskaya, A.A., Usmanova, I.N., Fedorenko, F.A.: Stochastic online optimization. Single-point and multi-point non-linear multi-armed bandits. convex and strongly-convex case. Autom. Remote Control 78(2), 224–234 (2017)

    Google Scholar 

  12. Goodfellow, I.: Nips 2016 tutorial: generative adversarial networks. arXiv preprint arXiv:1701.00160 (2016)

  13. Jin, Y., Sidford, A.: Efficiently solving MDPs with stochastic mirror descent. In: Daumé III, H., Singh, A. (eds.) Proceedings of the 37th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 119, pp. 4890–4900. PMLR, 13–18 July 2020

    Google Scholar 

  14. Nesterov, Y., Spokoiny, V.G.: Random gradient-free minimization of convex functions. Found. Comput. Math. 17(2), 527–566 (2017)

    CrossRef  MathSciNet  Google Scholar 

  15. Novitskii, V., Gasnikov, A.: Improved exploiting higher order smoothness in derivative-free optimization and continuous bandit. arXiv preprint arXiv:2101.03821 (2021)

  16. Sadiev, A., Beznosikov, A., Dvurechensky, P., Gasnikov, A.: Zeroth-order algorithms for smooth saddle-point problems. arXiv preprint arXiv:2009.09908 (2020)

  17. Shamir, O.: An optimal algorithm for bandit and zero-order convex optimization with two-point feedback. J. Mach. Learn. Res. 18(52), 1–11 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Zhang, Y., Zhou, Y., Ji, K., Zavlanos, M.M.: Improving the convergence rate of one-point zeroth-order optimization using residual feedback. arXiv preprint arXiv:2006.10820 (2020)

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Aleksandr Beznosikov .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Beznosikov, A., Novitskii, V., Gasnikov, A. (2021). One-Point Gradient-Free Methods for Smooth and Non-smooth Saddle-Point Problems. In: Pardalos, P., Khachay, M., Kazakov, A. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2021. Lecture Notes in Computer Science(), vol 12755. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-77875-0

  • Online ISBN: 978-3-030-77876-7

  • eBook Packages: Computer ScienceComputer Science (R0)