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arXiv · 2508.13569

Revisiting the Geometrically Decaying Step Size: Linear Convergence for Smooth or Non-Smooth Functions

Jihun Kim
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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.

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Jihun Kim. 2025-08-19. Revisiting the Geometrically Decaying Step Size: Linear Convergence for Smooth or Non-Smooth Functions. https://arxiv.org/abs/2508.13569

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