A Decentralized Proximal Gradient Tracking Algorithm for Composite Optimization on Riemannian Manifolds
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911056345956352 |
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| author | Wang, Lei Bao, Le Liu, Xin |
| author_facet | Wang, Lei Bao, Le Liu, Xin |
| contents | This paper focuses on minimizing a smooth function combined with a nonsmooth regularization term on a compact Riemannian submanifold embedded in the Euclidean space under a decentralized setting. Typically, there are two types of approaches at present for tackling such composite optimization problems. The first, subgradient-based approaches, rely on subgradient information of the objective function to update variables, achieving an iteration complexity of $\mathcal{O}(ε^{-4}\log^2(ε^{-2}))$. The second, smoothing approaches, involve constructing a smooth approximation of the nonsmooth regularization term, resulting in an iteration complexity of $\mathcal{O}(ε^{-4})$. This paper proposes a proximal gradient type algorithm that fully exploits the composite structure. The global convergence to a stationary point is established with a significantly improved iteration complexity of $\mathcal{O}(ε^{-2})$. To validate the effectiveness and efficiency of our proposed method, we present numerical results in real-world applications, showcasing its superior performance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11573 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Decentralized Proximal Gradient Tracking Algorithm for Composite Optimization on Riemannian Manifolds Wang, Lei Bao, Le Liu, Xin Optimization and Control This paper focuses on minimizing a smooth function combined with a nonsmooth regularization term on a compact Riemannian submanifold embedded in the Euclidean space under a decentralized setting. Typically, there are two types of approaches at present for tackling such composite optimization problems. The first, subgradient-based approaches, rely on subgradient information of the objective function to update variables, achieving an iteration complexity of $\mathcal{O}(ε^{-4}\log^2(ε^{-2}))$. The second, smoothing approaches, involve constructing a smooth approximation of the nonsmooth regularization term, resulting in an iteration complexity of $\mathcal{O}(ε^{-4})$. This paper proposes a proximal gradient type algorithm that fully exploits the composite structure. The global convergence to a stationary point is established with a significantly improved iteration complexity of $\mathcal{O}(ε^{-2})$. To validate the effectiveness and efficiency of our proposed method, we present numerical results in real-world applications, showcasing its superior performance. |
| title | A Decentralized Proximal Gradient Tracking Algorithm for Composite Optimization on Riemannian Manifolds |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2401.11573 |