Fundamental Benefit of Alternating Updates in Minimax Optimization
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916323929358336 |
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| author | Lee, Jaewook Cho, Hanseul Yun, Chulhee |
| author_facet | Lee, Jaewook Cho, Hanseul Yun, Chulhee |
| contents | The Gradient Descent-Ascent (GDA) algorithm, designed to solve minimax optimization problems, takes the descent and ascent steps either simultaneously (Sim-GDA) or alternately (Alt-GDA). While Alt-GDA is commonly observed to converge faster, the performance gap between the two is not yet well understood theoretically, especially in terms of global convergence rates. To address this theory-practice gap, we present fine-grained convergence analyses of both algorithms for strongly-convex-strongly-concave and Lipschitz-gradient objectives. Our new iteration complexity upper bound of Alt-GDA is strictly smaller than the lower bound of Sim-GDA; i.e., Alt-GDA is provably faster. Moreover, we propose Alternating-Extrapolation GDA (Alex-GDA), a general algorithmic framework that subsumes Sim-GDA and Alt-GDA, for which the main idea is to alternately take gradients from extrapolations of the iterates. We show that Alex-GDA satisfies a smaller iteration complexity bound, identical to that of the Extra-gradient method, while requiring less gradient computations. We also prove that Alex-GDA enjoys linear convergence for bilinear problems, for which both Sim-GDA and Alt-GDA fail to converge at all. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fundamental Benefit of Alternating Updates in Minimax Optimization Lee, Jaewook Cho, Hanseul Yun, Chulhee Optimization and Control Machine Learning The Gradient Descent-Ascent (GDA) algorithm, designed to solve minimax optimization problems, takes the descent and ascent steps either simultaneously (Sim-GDA) or alternately (Alt-GDA). While Alt-GDA is commonly observed to converge faster, the performance gap between the two is not yet well understood theoretically, especially in terms of global convergence rates. To address this theory-practice gap, we present fine-grained convergence analyses of both algorithms for strongly-convex-strongly-concave and Lipschitz-gradient objectives. Our new iteration complexity upper bound of Alt-GDA is strictly smaller than the lower bound of Sim-GDA; i.e., Alt-GDA is provably faster. Moreover, we propose Alternating-Extrapolation GDA (Alex-GDA), a general algorithmic framework that subsumes Sim-GDA and Alt-GDA, for which the main idea is to alternately take gradients from extrapolations of the iterates. We show that Alex-GDA satisfies a smaller iteration complexity bound, identical to that of the Extra-gradient method, while requiring less gradient computations. We also prove that Alex-GDA enjoys linear convergence for bilinear problems, for which both Sim-GDA and Alt-GDA fail to converge at all. |
| title | Fundamental Benefit of Alternating Updates in Minimax Optimization |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2402.10475 |