Abstract
In this paper, we consider two types of problems that have some similarity in their structure, namely, min-min problems and min-max saddle-point problems. Our approach is based on considering the outer minimization problem as a minimization problem with an inexact oracle. This inexact oracle is calculated via an inexact solution of the inner problem, which is either minimization or maximization problem. Our main assumption is that the available oracle is mixed: it is only possible to evaluate the gradient w.r.t. the outer block of variables which corresponds to the outer minimization problem, whereas for the inner problem, only zeroth-order oracle is available. To solve the inner problem, we use the accelerated gradient-free method with zeroth-order oracle. To solve the outer problem, we use either an inexact variant of Vaidya’s cutting-plane method or a variant of the accelerated gradient method. As a result, we propose a framework that leads to non-asymptotic complexity bounds for both min-min and min-max problems. Moreover, we estimate separately the number of first- and zeroth-order oracle calls, which are sufficient to reach any desired accuracy.
Keywords
- First-order methods
- Zeroth-order methods
- Cutting-plane methods
- Saddle-point problems
The research of A. Gasnikov and P. Dvurechensky was supported by Russian Science Foundation (project No. 21-71-30005). The research of E. Gladin, A. Sadiev and A. Beznosikov was partially supported by Andrei Raigorodskii scholarship.
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- 1.
Here and below instead of ARDDsc we can use Accelerated coordinate descent methods [9, 20] with replacing partial derivatives by finite differences. In this case we lost opportunity to play on the choice of the norm (that could save \(\sqrt{n_y}\)-factor in gradient-free oracle complexity estimate [6]), but, we gain a possibility to replace the wort case \(L_{yy}\) to the average one (that could be \(n_y\)-times smaller [20]). At the end this could also save \(\sqrt{n_y}\)-factor in gradient-free oracle complexity estimate [20].
- 2.
\(\widetilde{O} (\cdot )= O (\cdot )\) up to a small power of logarithmic factor.
References
Alkousa, M., Dvinskikh, D., Stonyakin, F., Gasnikov, A., Kovalev, D.: Accelerated methods for composite non-bilinear saddle point problem (2020). https://doi.org/10.1134/S0965542520110020
Beznosikov, A., Sadiev, A., Gasnikov, A.: Gradient-free methods for saddle-point problem. arXiv preprint arXiv:2005.05913 (2020).https://doi.org/10.1007/978-3-030-58657-7_11
Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision 40(1), 120–145 (2011)
Conn, A.R., Scheinberg, K., Vicente, L.N.: Introduction to derivative-free optimization. Soc. Ind. Appl. Math. (2009). https://doi.org/10.1137/1.9780898718768
Devolder, O., Glineur, F., Nesterov, Y.: First-order methods with inexact oracle: the strongly convex case (2013). http://hdl.handle.net/2078.1/128723
Dvurechensky, P., Gorbunov, E., Gasnikov, A.: An accelerated directional derivative method for smooth stochastic convex optimization. Eur. J. Oper. Res. 290(2), 601–621 (2021). https://doi.org/10.1016/j.ejor.2020.08.027
Fu, M.C. (ed.): Handbook of Simulation Optimization. ISORMS, vol. 216. Springer, New York (2015). https://doi.org/10.1007/978-1-4939-1384-8
Gasnikov, A., et al.: Universal method with inexact oracle and its applications for searching equillibriums in multistage transport problems. arXiv preprint arXiv:1506.00292 (2015)
Gasnikov, A., Dvurechensky, P., Usmanova, I.: On accelerated randomized methods. Proceedings of Moscow Institute of Physics and Technology 8, pp. 67–100. Russian (2016)
Goodfellow, I.J., et al.: Generative adversarial networks (2014)
Goodfellow, I.J., Shlens, J., Szegedy, C.: Explaining and harnessing adversarial examples (2014)
Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Eknomika i Matematicheskie Metody 12, 747–756 (1976)
Lin, H., Mairal, J., Harchaoui, Z.: A universal catalyst for first-order optimization. In: Cortes, C., Lawrence, N., Lee, D., Sugiyama, M., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 28. Curran Associates, Inc. (2015). https://proceedings.neurips.cc/paper/2015/file/c164bbc9d6c72a52c599bbb43d8db8e1-Paper.pdf
Liu, S., et al.: Min-max optimization without gradients: convergence and applications to adversarial ml (2019). http://proceedings.mlr.press/v119/liu20j.html
Madry, A., Makelov, A., Schmidt, L., Tsipras, D., Vladu, A.: Towards deep learning models resistant to adversarial attacks. In: 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, 30 April–3 May 2018, Conference Track Proceedings (2018)
Narodytska, N., Kasiviswanathan, S.P.: Simple black-box adversarial attacks on deep neural networks. In: CVPR Workshops. pp. 1310–1318. IEEE Computer Society (2017). http://doi.ieeecomputersociety.org/10.1109/CVPRW.2017.172
Nedić, A., Ozdaglar, A.: Subgradient methods for saddle-point problems. J. Optim. Theory Appl. 142(1), 205–228 (2009)
Nemirovski, A.: Prox-method with rate of convergence o (1/ t ) for variational inequalities with lipschitz continuous monotone operators and smooth convex-concave saddle point problems. SIAM J. Optim. 15, 229–251 (2004). https://doi.org/10.1137/S1052623403425629
Nesterov, Y.: Lectures on Convex Optimization. SOIA, vol. 137. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-91578-4
Nesterov, Y., Stich, S.U.: Efficiency of the accelerated coordinate descent method on structured optimization problems. SIAM J. Optim. 27(1), 110–123 (2017)
Pinto, L., Davidson, J., Sukthankar, R., Gupta, A.: Robust adversarial reinforcement learning. Proceedings of Machine Learning Research, 06–11 August 2017, vol. 70, pp. 2817–2826. PMLR, International Convention Centre, Sydney (2017). http://proceedings.mlr.press/v70/pinto17a.html
Polyak, B.T.: Introduction to Optimization. Publications Division, Inc., New York (1987)
Sadiev, A., Beznosikov, A., Dvurechensky, P., Gasnikov, A.: Zeroth-order algorithms for smooth saddle-point problems. arXiv:2009.09908 (2020)
Shashaani, S., Hashemi, F.S., Pasupathy, R.: Astro-df: a class of adaptive sampling trust-region algorithms for derivative-free stochastic optimization. SIAM J. Optim. 28(4), 3145–3176 (2018). https://doi.org/10.1137/15M1042425
Stonyakin, F., et al.: Inexact relative smoothness and strong convexity for optimization and variational inequalities by inexact model (2020). https://doi.org/10.1080/10556788.2021.1924714
Tramèr, F., Kurakin, A., Papernot, N., Goodfellow, I., Boneh, D., McDaniel, P.: Ensemble adversarial training: attacks and defenses (2017). https://openreview.net/forum?id=rkZvSe-RZ
Vaidya, P.M.: A new algorithm for minimizing convex functions over convex sets. In: 30th Annual Symposium on Foundations of Computer Science, pp. 338–343. IEEE Computer Society (1989)
Vaidya, P.M.: A new algorithm for minimizing convex functions over convex sets. Math. Program. 73(3), 291–341 (1996)
Wang, Z., Balasubramanian, K., Ma, S., Razaviyayn, M.: Zeroth-order algorithms for nonconvex minimax problems with improved complexities (2020)
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Gladin, E., Sadiev, A., Gasnikov, A., Dvurechensky, P., Beznosikov, A., Alkousa, M. (2021). Solving Smooth Min-Min and Min-Max Problems by Mixed Oracle Algorithms. In: Strekalovsky, A., Kochetov, Y., Gruzdeva, T., Orlov, A. (eds) Mathematical Optimization Theory and Operations Research: Recent Trends. MOTOR 2021. Communications in Computer and Information Science, vol 1476. Springer, Cham. https://doi.org/10.1007/978-3-030-86433-0_2
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