A Retraction-Free EXTRA Method for Decentralized Optimization on the Stiefel Manifold
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908996796940288 |
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| author | Li, Shu Hu, Jiang |
| author_facet | Li, Shu Hu, Jiang |
| contents | Decentralized optimization provides a fundamental framework for large-scale learning and signal processing with distributed data. We study decentralized optimization with orthogonality constraints on the Stiefel manifold and propose RF-EXTRA, a distributed retraction-free primal-dual method on static undirected networks. The method combines an approximate gradient mapping for orthogonality-constrained optimization with an EXTRA-based decentralized recursion, thereby avoiding retractions while preserving a simple communication pattern. On the theoretical side, the analysis considers \revise{the joint error} $(\mathbf{X}_k-\overline{\mathbf X}_k,\mathbf{s}_k-\overline{\mathbf s}_k)$ in the local variables and local directions, and establishes a contractive recursion for the joint error. This contractivity ensures that the joint error can be controlled using small yet constant step sizes, thus leading to an exact $\mathcal{O}(1/K)$ convergence rate of RF-EXTRA to a stationary point. Experiments on PCA and low-rank matrix completion show that RF-EXTRA compares favorably with the reported decentralized baselines and exhibits strong communication efficiency on the tested tasks on the Stiefel manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23754 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Retraction-Free EXTRA Method for Decentralized Optimization on the Stiefel Manifold Li, Shu Hu, Jiang Optimization and Control Decentralized optimization provides a fundamental framework for large-scale learning and signal processing with distributed data. We study decentralized optimization with orthogonality constraints on the Stiefel manifold and propose RF-EXTRA, a distributed retraction-free primal-dual method on static undirected networks. The method combines an approximate gradient mapping for orthogonality-constrained optimization with an EXTRA-based decentralized recursion, thereby avoiding retractions while preserving a simple communication pattern. On the theoretical side, the analysis considers \revise{the joint error} $(\mathbf{X}_k-\overline{\mathbf X}_k,\mathbf{s}_k-\overline{\mathbf s}_k)$ in the local variables and local directions, and establishes a contractive recursion for the joint error. This contractivity ensures that the joint error can be controlled using small yet constant step sizes, thus leading to an exact $\mathcal{O}(1/K)$ convergence rate of RF-EXTRA to a stationary point. Experiments on PCA and low-rank matrix completion show that RF-EXTRA compares favorably with the reported decentralized baselines and exhibits strong communication efficiency on the tested tasks on the Stiefel manifold. |
| title | A Retraction-Free EXTRA Method for Decentralized Optimization on the Stiefel Manifold |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2604.23754 |