Optimal Subgradient Methods for Lipschitz Convex Optimization with Error Bounds

Alex L. Wang

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arXiv · 2512.13863

Optimal Subgradient Methods for Lipschitz Convex Optimization with Error Bounds

Alex L. Wang

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Abstract

We study the iteration complexity of Lipschitz convex optimization problems satisfying a general error bound. We show that for this class of problems, subgradient descent with either Polyak stepsizes or decaying stepsizes achieves minimax optimal convergence guarantees for decreasing distance-to-optimality. The main contribution is a novel lower-bounding argument that produces hard functions simultaneously satisfying zero-chain conditions and global error bounds.

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Alex L. Wang. 2025-12-15. Optimal Subgradient Methods for Lipschitz Convex Optimization with Error Bounds. https://arxiv.org/abs/2512.13863

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