An Alternating Gradient Projection Algorithm with Momentum for Nonconvex-Concave Minimax Problems

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  • 1 School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China;
    2 Big Data & Intelligence Engineering School, Chongqing College of International Business and Economics, Chongqing 401520, China

Received date: 2023-05-09

  Revised date: 2024-03-05

  Online published: 2026-07-06

Supported by

This work was partially supported by the National Key R&D Program of China (No. 2023YFA1011303), the National Natural Science Foundation of China (Nos. 11971083 and 11991024), the Team Project of Innovation Leading Talent in Chongqing (No. CQYC20210309536), and the Contract System Project of Chongqing Talent Plan (No. cstc2022ycjh-bgzxm0147).

Abstract

The growing interest in addressing minimax optimization problem has been fueled by recent applications in machine learning. Although extensively studied in the convex-concave regime, where a global solution can be efficiently computed, this paper delves into the minimax problem within the nonconvex-concave setup. We propose an alternating gradient projection algorithm with momentum (M-AGP), belonging to single-loop algorithms that not only are easier to implement but also require only the computation of gradient projection updates. We demonstrate that the proposed algorithm identifies an ε-stationary point of the nonconvex-strongly concave minimax problem in O(ε-2) iterations, representing the best-known rate in the literature. Finally, we utilize two test problems, namely robust nonlinear regression and an image classification problem, to showcase the efficacy of the proposed algorithm.

Cite this article

Jue-You Li, Tao Xie . An Alternating Gradient Projection Algorithm with Momentum for Nonconvex-Concave Minimax Problems[J]. Journal of the Operations Research Society of China, 2026 , 14(2) : 632 -652 . DOI: 10.1007/s40305-024-00540-5

References

[1] Balduzzi, D., Racaniere, S., Martens, J., Foerster, J., Tuyls, K., Graepel, T.: The mechanics of n-player differentiable games. In: International Conference on Machine Learning, PMLR, pp. 354–363(2018)
[2] Chen, Y., Lan, G., Ouyang, Y.: Optimal primal-dual methods for a class of saddle point problems. SIAM J. Optim. 24, 1779–1814(2014)
[3] Chen, Y., Lan, G., Ouyang, Y.: Accelerated schemes for a class of variational inequalities. Math. Program. 165, 113–149(2017)
[4] Creswell, A., White, T., Dumoulin, V., Arulkumaran, K., Sengupta, B., Bharath, A.: Generative adversarial networks: an overview. IEEE Signal Proc. Mag. 35, 53–65(2018)
[5] Chan, E., Lin, C., Chan, M., Nagano, M., Pan, B.: Efficient geometry-aware 3D generative adversarial networks. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 16123–16133(2022)
[6] Daskalakis,C.,Panageas,I.:Thelimitpointsof (optimistic) gradientdescentin min–max optimization. In: Advances in Neural Information Processing Systems, vol. 31, pp. 1–11(2018)
[7] Dai, Y.H., Zhang, L.W.: Optimality conditions for constrained minimax optimization. CSIAM Trans. Appl. Math. 1, 296–315(2020)
[8] Dai, Y.H., Zhang, L.W.: The rate of convergence of augmented Lagrangian method for minimax optimization problems with equality constraints. J. Oper. Res. Soc. China (2022). https://doi.org/10. 1007/s40305-022-00439-z
[9] Gidel, G., Hemmat, R., Pezeshki, M., Priol, R., Huang, G. Julien, S., Mitliagkas, I.: Negative momentum for improved game dynamics. In: The 22nd International Conference on Artificial Intelligence and Statistics, PMLR, pp. 1802–1811(2019)
[10] Juditsky, A., Nemirovski, A.: Solving variational inequalities with monotone operators on domains given by linear minimization oracles. Math. Program. 156, 221–256(2016)
[11] Letcher, A., Balduzzi, D., Racaniere, S., Martens, J., Foerster, J., Tuyls, K., Graepel, T.: Differentiable game mechanics. J. Mach. Learn. Res. 20, 1–40(2019)
[12] Lin, T., Jin, C., Jordan, M.: Near-optimal algorithms for minimax optimization. In: Conference on Learning Theory, PMLR, pp. 2738–2779(2020)
[13] Lin, T., Jin, C., Jordan, M.: On gradient descent ascent for nonconvex–concave minimax problems. In: International Conference on Machine Learning, PMLR, pp. 6083–6093(2020)
[14] Lu, S., Tsaknakis, I., Hong, M., Chen, Y.: Hybrid block successive approximation for onesided nonconvex min-max problems: algorithms and applications. IEEE Signal Proc. Mag. 68, 3676–3691(2021)
[15] Mescheder, L., Geiger, A., Nowozin, S.: Which training methods for GANs do actually converge? In: International Conference on Machine Learning, PMLR, pp. 3481–3490(2018)
[16] Mai, T., Mihail, M., Panageas, I., Ratcliff, W., Vazirani, V., Yunker, P.: Cycles in zero-sum differential games and biological diversity. In: Proceedings of the 2018 ACM Conference on Economics and Computation, pp. 339–350(2018)
[17] Mokhtari, A., Ozdaglar, A., Pattathil, S.: Convergence rate of O(1/k) for optimistic gradient and extragradient methods in smooth convex-concave saddle point problems. SIAM J. Optim. 30, 3230– 3251(2020)
[18] Nesterov, Y.: Dual extrapolation and its applications to solving variational inequalities and related problems. Math. Program. 109, 319–344(2007)
[19] 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)
[20] Nemirovski, A., Juditsky, A., Lan, G.: Robust stochastic approximation approach to stochastic programming. SIAM J. Optim. 19, 1574–1609(2009)
[21] Nouiehed, M., Sanjabi, M., Huang, T., Lee, J., Razaviyayn, M.: Solving a class of non-convex min–max games using iterative first order methods. In: Advances in Neural Information Processing Systems, vol. 32, pp. 311–319(2019)
[22] Rafique, H., Liu, M., Lin, Q., Yang, T.: Weakly-convex-concave min-max optimization: provable algorithms and applications in machine learning. Optim. Method Softw. 37, 1087–1121(2022)
[23] Shen, J., Wang, Z., Xu, Z.: Zeroth-order single-loop algorithms for nonconvex-linear minimax problems. J. Global Optim. 87, 551–580(2023)
[24] Thekumparampil, K., Jain, P., Netrapalli, P., Oh, S.: Efficient algorithms for smooth minimax optimization. In: Advances in Neural Information Processing Systems, vol. 32, pp. 1–10(2019)
[25] Xu, Z., Zhang, H.: Optimization algorithms and their complexity analysis for non-convex minimax problems. Oper. Res. Trans. 25, 74–86(2021). (in Chinese)
[26] Xu, Z., Shen, J., Wang, Z., Dai, Y.: Zeroth-order alternating randomized gradient projection algorithms for general nonconvex–concave minimax problems (2021). arXiv:2108.00473
[27] Xu, Z., Zhang, H., Xu, Y., Lan, G.: A unified single-loop alternating gradient projection algorithm for nonconvex-concave and convex-nonconcave minimax problems. Math. Program. 201, 635–706(2023)
[28] Yang, J., Orvieto, A., Lucchi, A., He, N.: Faster single-loop algorithms for minimax optimization without strong concavity. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp. 5485–5517(2022)
[29] Zhang, J., Xiao, P., Sun, R., Luo, Z.: A single-loop smoothed gradient descent–ascent algorithm for nonconvex–concave min–max problems. In: Advances in Neural Information Processing Systems, vol. 33, pp. 7377–7389(2020)
[30] Zhang, H., Xu, Y., Xu, Z.: Block alternating proximal gradient algorithm for convex–nonconcave minimax problems. Oper. Res. Trans. 26, 65–73(2022). (in Chinese)
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