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arXiv · 2607.19240

A Complete Characterization of Optimal Subgradient Methods for Lipschitz Convex Minimization

Aaron Zoll·Benjamin Grimmer
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Abstract

We consider the design of optimal fixed-step first-order methods for $M$-Lipschitz convex optimization given $\|x_0-x_\star\|\leq D$. Prior works have identified several distinct fixed-step methods, parameterized by a matrix of stepsizes $W$, with the (information-theoretic) minimax optimal rate $MD/\sqrt{N+1}$ of objective gap convergence. We provide a complete characterization of every optimal fixed-step method. Moreover, we show every optimal fixed-step method can be derived from the constructive approach of~\cite{constructive_approach} and provide a polyhedral representation of the set of optimal methods through proof multipliers. From this characterization, we show that no anytime optimal fixed-step subgradient methods exist.

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Aaron Zoll, Benjamin Grimmer. 2026-07-21. A Complete Characterization of Optimal Subgradient Methods for Lipschitz Convex Minimization. https://arxiv.org/abs/2607.19240

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