Abstract
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.
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, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Akhavan, A., Pontil, M., Tsybakov, A.B.: Exploiting higher order smoothness in derivative-free optimization and continuous bandits. arXiv preprint arXiv:2006.07862 (2020)
Bach, F., Perchet, V.: Highly-smooth zero-th order online optimization. In: Conference on Learning Theory, pp. 257–283. PMLR (2016)
Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization: analysis, algorithms, and engineering applications (2019)
Beznosikov, A., Gorbunov, E., Gasnikov, A.: Derivative-free method for composite optimization with applications to decentralized distributed optimization. arXiv preprint arXiv:1911.10645 (2019)
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)
Beznosikov, A., Sadiev, A., Gasnikov, A.: Gradient-free methods for saddle-point problem. arXiv preprint arXiv:2005.05913 (2020)
Carmon, Y., Jin, Y., Sidford, A., Tian, K.: Coordinate methods for matrix games. arXiv preprint arXiv:2009.08447 (2020)
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). https://doi.org/10.1145/3128572.3140448. http://dx.doi.org/10.1145/3128572.3140448
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)
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)
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)
Goodfellow, I.: Nips 2016 tutorial: generative adversarial networks. arXiv preprint arXiv:1701.00160 (2016)
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
Nesterov, Y., Spokoiny, V.G.: Random gradient-free minimization of convex functions. Found. Comput. Math. 17(2), 527–566 (2017)
Novitskii, V., Gasnikov, A.: Improved exploiting higher order smoothness in derivative-free optimization and continuous bandit. arXiv preprint arXiv:2101.03821 (2021)
Sadiev, A., Beznosikov, A., Dvurechensky, P., Gasnikov, A.: Zeroth-order algorithms for smooth saddle-point problems. arXiv preprint arXiv:2009.09908 (2020)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
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. https://doi.org/10.1007/978-3-030-77876-7_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-77876-7_10
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)