Proximal Gradient Descent Ascent Methods for Nonsmooth Nonconvex-Concave Minimax Problems on Riemannian Manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908903635156992 |
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| author | Xie, Xiyuan Li, Qia |
| author_facet | Xie, Xiyuan Li, Qia |
| contents | Nonsmooth nonconvex-concave minimax problems have attracted significant attention due to their wide applications in many fields. In this paper, we consider a class of nonsmooth nonconvex-concave minimax problems on Riemannian manifolds. Owing to the nonsmoothness of the objective function, existing minimax manifold optimization methods cannot be directly applied to solve this problem. We propose two manifold proximal gradient descent ascent (MPGDA) algorithms for solving the problem. The first algorithm alternatively performs one or multiple manifold proximal gradient descent steps and a proximal ascent step at each iteration, and we prove that it can find an $\varepsilon$-game-stationary point and an $\varepsilon$-optimization-stationary point within $\mathcal{O}(\varepsilon^{-3})$ outer iterations. The second algorithm alternatively performs one manifold proximal gradient descent step and a proximal gradient ascent step, and we show that it can reach an $\varepsilon$-game-stationary point and an $\varepsilon$-optimization-stationary point within $\mathcal{O}(\varepsilon^{-4})$ outer iterations. Numerical experiments on an analytic example, fair sparse PCA, and sparse spectral clustering are conducted to illustrate the advantages of the proposed algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02140 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proximal Gradient Descent Ascent Methods for Nonsmooth Nonconvex-Concave Minimax Problems on Riemannian Manifolds Xie, Xiyuan Li, Qia Optimization and Control 90C26 and 90C30 and 65K05 Nonsmooth nonconvex-concave minimax problems have attracted significant attention due to their wide applications in many fields. In this paper, we consider a class of nonsmooth nonconvex-concave minimax problems on Riemannian manifolds. Owing to the nonsmoothness of the objective function, existing minimax manifold optimization methods cannot be directly applied to solve this problem. We propose two manifold proximal gradient descent ascent (MPGDA) algorithms for solving the problem. The first algorithm alternatively performs one or multiple manifold proximal gradient descent steps and a proximal ascent step at each iteration, and we prove that it can find an $\varepsilon$-game-stationary point and an $\varepsilon$-optimization-stationary point within $\mathcal{O}(\varepsilon^{-3})$ outer iterations. The second algorithm alternatively performs one manifold proximal gradient descent step and a proximal gradient ascent step, and we show that it can reach an $\varepsilon$-game-stationary point and an $\varepsilon$-optimization-stationary point within $\mathcal{O}(\varepsilon^{-4})$ outer iterations. Numerical experiments on an analytic example, fair sparse PCA, and sparse spectral clustering are conducted to illustrate the advantages of the proposed algorithms. |
| title | Proximal Gradient Descent Ascent Methods for Nonsmooth Nonconvex-Concave Minimax Problems on Riemannian Manifolds |
| topic | Optimization and Control 90C26 and 90C30 and 65K05 |
| url | https://arxiv.org/abs/2505.02140 |