{"ID":2880556,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.13569","arxiv_id":"2508.13569","title":"Revisiting the Geometrically Decaying Step Size: Linear Convergence for Smooth or Non-Smooth Functions","abstract":"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.","short_abstract":"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 weak...","url_abs":"https://arxiv.org/abs/2508.13569","url_pdf":"https://arxiv.org/pdf/2508.13569v3","authors":"[\"Jihun Kim\"]","published":"2025-08-19T07:00:01Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}