Revisiting the Geometrically Decaying Step Size: Linear Convergence for Smooth or Non-Smooth Functions
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917120605945856 |
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| author | Kim, Jihun |
| author_facet | Kim, Jihun |
| contents | We revisit the geometrically decaying step size given a positive inverse condition number, under which a locally Lipschitz function shows linear convergence. The positivity does not require the function to satisfy convexity, weak convexity, quasar convexity, or sharpness, but instead amounts to a property strictly weaker than the assumptions used in existing works (e.g., weak convexity + sharpness). We propose a clean and simple subgradient descent algorithm that requires minimal knowledge of problem constants, applicable to either smooth or non-smooth functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13569 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Revisiting the Geometrically Decaying Step Size: Linear Convergence for Smooth or Non-Smooth Functions Kim, Jihun Optimization and Control 49J52, 90C26, 90C30, 90C56 We revisit the geometrically decaying step size given a positive inverse condition number, under which a locally Lipschitz function shows linear convergence. The positivity does not require the function to satisfy convexity, weak convexity, quasar convexity, or sharpness, but instead amounts to a property strictly weaker than the assumptions used in existing works (e.g., weak convexity + sharpness). We propose a clean and simple subgradient descent algorithm that requires minimal knowledge of problem constants, applicable to either smooth or non-smooth functions. |
| title | Revisiting the Geometrically Decaying Step Size: Linear Convergence for Smooth or Non-Smooth Functions |
| topic | Optimization and Control 49J52, 90C26, 90C30, 90C56 |
| url | https://arxiv.org/abs/2508.13569 |