Revisit First-order Methods for Geodesically Convex Optimization
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918161361666048 |
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| author | Shu, Yunlu Jiang, Jiaxin Shi, Lei Wang, Tianyu |
| author_facet | Shu, Yunlu Jiang, Jiaxin Shi, Lei Wang, Tianyu |
| contents | In a seminal work of Zhang and Sra, gradient descent methods for geodesically convex optimization were comprehensively studied. In particular, Zhang and Sra derived a comparison inequality that relates the iterative points in the optimization process. Since their seminal work, numerous follow-ups have studied different downstream usages of their comparison lemma.
In this work, we introduce the concept of quasilinearization to optimization, presenting a novel framework for analyzing geodesically convex optimization. By leveraging this technique, we establish state-of-the-art convergence rates -- for both deterministic and stochastic settings -- under weaker assumptions than previously required. The technique of quasilinearization may prove valuable for other non-Euclidean optimization problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06814 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Revisit First-order Methods for Geodesically Convex Optimization Shu, Yunlu Jiang, Jiaxin Shi, Lei Wang, Tianyu Optimization and Control In a seminal work of Zhang and Sra, gradient descent methods for geodesically convex optimization were comprehensively studied. In particular, Zhang and Sra derived a comparison inequality that relates the iterative points in the optimization process. Since their seminal work, numerous follow-ups have studied different downstream usages of their comparison lemma. In this work, we introduce the concept of quasilinearization to optimization, presenting a novel framework for analyzing geodesically convex optimization. By leveraging this technique, we establish state-of-the-art convergence rates -- for both deterministic and stochastic settings -- under weaker assumptions than previously required. The technique of quasilinearization may prove valuable for other non-Euclidean optimization problems. |
| title | Revisit First-order Methods for Geodesically Convex Optimization |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2504.06814 |