An approximation proximal gradient algorithm for nonconvex-linear minimax problems with nonconvex nonsmooth terms
Nonconvex minimax problems have attracted significant attention in machine learning, wireless communication and many other fields. In this paper, we propose an efficient approximation proximal gradient algorithm for solving a class of nonsmooth nonconvex-linear minimax problems with a nonconvex nons...
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| Veröffentlicht in: | Journal of global optimization Jg. 90; H. 1; S. 73 - 92 |
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| Abstract | Nonconvex minimax problems have attracted significant attention in machine learning, wireless communication and many other fields. In this paper, we propose an efficient approximation proximal gradient algorithm for solving a class of nonsmooth nonconvex-linear minimax problems with a nonconvex nonsmooth term, and the number of iteration to find an
ε
-stationary point is upper bounded by
O
(
ε
-
3
)
. Some numerical results on one-bit precoding problem in massive MIMO system and a distributed non-convex optimization problem demonstrate the effectiveness of the proposed algorithm. |
|---|---|
| AbstractList | Nonconvex minimax problems have attracted significant attention in machine learning, wireless communication and many other fields. In this paper, we propose an efficient approximation proximal gradient algorithm for solving a class of nonsmooth nonconvex-linear minimax problems with a nonconvex nonsmooth term, and the number of iteration to find an ε-stationary point is upper bounded by O(ε-3). Some numerical results on one-bit precoding problem in massive MIMO system and a distributed non-convex optimization problem demonstrate the effectiveness of the proposed algorithm. Nonconvex minimax problems have attracted significant attention in machine learning, wireless communication and many other fields. In this paper, we propose an efficient approximation proximal gradient algorithm for solving a class of nonsmooth nonconvex-linear minimax problems with a nonconvex nonsmooth term, and the number of iteration to find an [Formula omitted]-stationary point is upper bounded by [Formula omitted]. Some numerical results on one-bit precoding problem in massive MIMO system and a distributed non-convex optimization problem demonstrate the effectiveness of the proposed algorithm. Nonconvex minimax problems have attracted significant attention in machine learning, wireless communication and many other fields. In this paper, we propose an efficient approximation proximal gradient algorithm for solving a class of nonsmooth nonconvex-linear minimax problems with a nonconvex nonsmooth term, and the number of iteration to find an ε -stationary point is upper bounded by O ( ε - 3 ) . Some numerical results on one-bit precoding problem in massive MIMO system and a distributed non-convex optimization problem demonstrate the effectiveness of the proposed algorithm. |
| Audience | Academic |
| Author | He, Jiefei Zhang, Huiling Xu, Zi |
| Author_xml | – sequence: 1 givenname: Jiefei surname: He fullname: He, Jiefei organization: Department of Mathematics, College of Sciences, Shanghai University – sequence: 2 givenname: Huiling surname: Zhang fullname: Zhang, Huiling organization: Department of Mathematics, College of Sciences, Shanghai University – sequence: 3 givenname: Zi orcidid: 0000-0003-0968-8027 surname: Xu fullname: Xu, Zi email: xuzi@shu.edu.cn organization: Department of Mathematics, College of Sciences, Shanghai University, Newtouch Center for Mathematics of Shanghai University, Shanghai University |
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| Cites_doi | 10.1109/TSP.2020.2986363 10.1109/ICASSP.2019.8683795 10.1109/TSP.2008.919636 10.1007/s10107-015-0957-3 10.1080/10556788.2021.1895152 10.1007/s10898-022-01169-5 10.1109/TWC.2018.2868369 10.1007/s10107-019-01365-4 10.1007/s10107-022-01919-z 10.1137/20M1337600 10.1137/20M1313222 10.1007/s10589-020-00237-4 |
| ContentType | Journal Article |
| Copyright | The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. COPYRIGHT 2024 Springer |
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| Keywords | Nonsmooth problem Iteration complexity 90C30 90C47 90C26 Nonconvex-linear minimax problem Approximation proximal gradient algorithm |
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| References | ShenJWangZXuZZeroth-order single-loop algorithms for nonconvex-linear minimax problemsJ. Glob. Optim.2023872551580466181310.1007/s10898-022-01169-5 YildizMEScaglioneACoding with side information for rate-constrained consensusIEEE Trans. Signal Process.200856837533764251705910.1109/TSP.2008.919636 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. Progr., Ser. A (2023). https://doi.org/10.1007/s10107-022-01919-z ThekumparampilKKJainPNetrapalliPOhSEfficient algorithms for smooth minimax optimizationAdv. Neural Inf. Process. Syst.2019321265912670 Nouiehed, M., Sanjabi, M., Huang, T., Lee, J.D., Razaviyayn, M.: Solving a class of non-convex min-max games using iterative first order methods. In: Advances in Neural Information Processing Systems (2019) YangJZhangSKiyavashNHeNA catalyst framework for minimax optimizationAdv. Neural. Inf. Process. 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In: International Conference on Learning Representations (2018) LuSTsaknakisIHongMChenYHybrid block successive approximation for one-sided non-convex min-max problems: algorithms and applicationsIEEE Trans. Signal Process.20206836763691412810610.1109/TSP.2020.2986363 Wu, Z., Jiang, B., Liu, Y.F., Dai, Y.H.: CI-based one-bit precoding for multiuser downlink massive MIMO systems with PSK modulation: a negative ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1 $$\end{document} penalty approach (2021). arXiv preprint arXiv:2110.11628 Gidel, G., Berard, H., Vignoud, G., Vincent, P., Lacoste-Julien, S.: A variational inequality perspective on generative adversarial networks. 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Splitting methods in communication, imaging, science, and engineering2017ChamSpringer461497 M Asteris (1383_CR30) 2014; 32 T Lin (1383_CR21) 2020; 119 1383_CR3 1383_CR10 1383_CR6 1383_CR5 1383_CR15 C Jin (1383_CR24) 2020; 119 DM Ostrovskii (1383_CR11) 2021; 31 1383_CR18 J Shen (1383_CR29) 2023; 87 S Lu (1383_CR25) 2020; 68 GB Giannakis (1383_CR1) 2017 ME Yildiz (1383_CR31) 2008; 56 W Kong (1383_CR8) 2021; 31 J Yang (1383_CR14) 2020; 33 1383_CR22 W Pan (1383_CR28) 2021; 78 1383_CR26 KK Thekumparampil (1383_CR13) 2019; 32 Z Xu (1383_CR23) 2023; 24 D Hajinezhad (1383_CR2) 2019; 176 A Letcher (1383_CR20) 2019; 20 A Li (1383_CR4) 2018; 17 J Ho (1383_CR19) 2016; 29 J Zhang (1383_CR27) 2020; 33 H Rafique (1383_CR12) 2022; 37 C Daskalakis (1383_CR16) 2018; 31 T Lin (1383_CR9) 2020; 125 Q Qian (1383_CR7) 2019; 33 A Chambolle (1383_CR17) 2016; 159 |
| References_xml | – reference: YangJZhangSKiyavashNHeNA catalyst framework for minimax optimizationAdv. Neural. Inf. Process. Syst.20203356675678 – reference: Daskalakis, C., Ilyas, A., Syrgkanis, V., Zeng, H.: Training GANs with optimism. In: International Conference on Learning Representations (2018) – reference: Lu, S., Tsaknakis, I., Hong, M.: Block alternating optimization for non-convex min-max problems: algorithms and applications in signal processing and communications. In: ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 4754–4758 (2019) – reference: HajinezhadDHongMPerturbed proximal primal-dual algorithm for nonconvex nonsmooth optimizationMath. Program.20191761–2207245396080910.1007/s10107-019-01365-4 – reference: JinCNetrapalliPJordanMWhat is local optimality in nonconvex-nonconcave minimax optimization?Int. Confer. Mach. Learn.202011948804889 – reference: PanWShenJXuZAn efficient algorithm for nonconvex-linear minimax optimization problem and its application in solving weighted maximin dispersion problemComput. Optim. Appl.202178287306419873410.1007/s10589-020-00237-4 – reference: HoJErmonSGenerative adversarial imitation learningAdv. Neural Inf. Process. Syst.20162945654573 – reference: LuSTsaknakisIHongMChenYHybrid block successive approximation for one-sided non-convex min-max problems: algorithms and applicationsIEEE Trans. Signal Process.20206836763691412810610.1109/TSP.2020.2986363 – reference: ThekumparampilKKJainPNetrapalliPOhSEfficient algorithms for smooth minimax optimizationAdv. Neural Inf. Process. Syst.2019321265912670 – reference: YildizMEScaglioneACoding with side information for rate-constrained consensusIEEE Trans. Signal Process.200856837533764251705910.1109/TSP.2008.919636 – reference: Xu, Z., Shen, J., Wang, Z., Dai, Y.H.: Derivative-free Alternating Projection Algorithms for General Nonconvex-Concave Minimax Problems. SIAM Journal on Optimization, accepted, arXiv preprint arXiv:2108.00473 – reference: Nouiehed, M., Sanjabi, M., Huang, T., Lee, J.D., Razaviyayn, M.: Solving a class of non-convex min-max games using iterative first order methods. In: Advances in Neural Information Processing Systems (2019) – reference: LinTJinCJordanMOn gradient descent ascent for nonconvex-concave minimax problemsInt. Confer. Mach. Learn.202011960836093 – reference: LiAMasourosCLiuFSwindlehurstALMassive MIMO 1-bit DAC transmission: a low-complexity symbol scaling approachIEEE Trans. Wirel. Commun.201817117559757510.1109/TWC.2018.2868369 – reference: DaskalakisCPanageasIThe limit points of (optimistic) gradient descent in min-max optimizationAdv. Neural Inf. Process. 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| SubjectTerms | Algorithms Approximation Computer Science Convex analysis Convexity Machine learning Mathematics Mathematics and Statistics MIMO communications Minimax technique Operations Research/Decision Theory Optimization Real Functions Wireless communications |
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| Title | An approximation proximal gradient algorithm for nonconvex-linear minimax problems with nonconvex nonsmooth terms |
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