Stochastic Approximation Proximal Subgradient Method for Stochastic Convex-Concave Minimax Optimization
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929296273047552 |
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| author | Dai, Yu-Hong Wang, Jiani Zhang, Liwei |
| author_facet | Dai, Yu-Hong Wang, Jiani Zhang, Liwei |
| contents | This paper presents a stochastic approximation proximal subgradient (SAPS) method for stochastic convex-concave minimax optimization. By accessing unbiased and variance bounded approximate subgradients, we show that this algorithm exhibits ${\rm O}(N^{-1/2})$ expected convergence rate of the minimax optimality measure if the parameters in the algorithm are properly chosen, where $N$ denotes the number of iterations. Moreover, we show that the algorithm has ${\rm O}(\log(N)N^{-1/2})$ minimax optimality measure bound with high probability. Further we study a specific stochastic convex-concave minimax optimization problems arising from stochastic convex conic optimization problems, which the the bounded subgradient condition is fail. To overcome the lack of the bounded subgradient conditions in convex-concave minimax problems, we propose a linearized stochastic approximation augmented Lagrange (LSAAL) method and prove that this algorithm exhibits ${\rm O}(N^{-1/2})$ expected convergence rate for the minimax optimality measure and ${\rm O}(\log^2(N)N^{-1/2})$ minimax optimality measure bound with high probability as well. Preliminary numerical results demonstrate the effect of the SAPS and LSAAL methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_20205 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stochastic Approximation Proximal Subgradient Method for Stochastic Convex-Concave Minimax Optimization Dai, Yu-Hong Wang, Jiani Zhang, Liwei Optimization and Control 90C30 This paper presents a stochastic approximation proximal subgradient (SAPS) method for stochastic convex-concave minimax optimization. By accessing unbiased and variance bounded approximate subgradients, we show that this algorithm exhibits ${\rm O}(N^{-1/2})$ expected convergence rate of the minimax optimality measure if the parameters in the algorithm are properly chosen, where $N$ denotes the number of iterations. Moreover, we show that the algorithm has ${\rm O}(\log(N)N^{-1/2})$ minimax optimality measure bound with high probability. Further we study a specific stochastic convex-concave minimax optimization problems arising from stochastic convex conic optimization problems, which the the bounded subgradient condition is fail. To overcome the lack of the bounded subgradient conditions in convex-concave minimax problems, we propose a linearized stochastic approximation augmented Lagrange (LSAAL) method and prove that this algorithm exhibits ${\rm O}(N^{-1/2})$ expected convergence rate for the minimax optimality measure and ${\rm O}(\log^2(N)N^{-1/2})$ minimax optimality measure bound with high probability as well. Preliminary numerical results demonstrate the effect of the SAPS and LSAAL methods. |
| title | Stochastic Approximation Proximal Subgradient Method for Stochastic Convex-Concave Minimax Optimization |
| topic | Optimization and Control 90C30 |
| url | https://arxiv.org/abs/2403.20205 |