On Minimax Optimality and Uniqueness of Fixed-Step First-Order Methods for Smooth Convex Optimization
Abstract
This paper considers the design of optimal fixed-step first-order methods for high-dimensional minimization of L-smooth convex functions. For optimizing worst-case performance measured via suboptimality of the final function value (relative to the initial squared distance to a minimizer), we provide an algebraic proof of the optimality of the optimized gradient method (OGM) and establish its uniqueness among all fixed-step first-order methods. For the alternative measure of final squared gradient norm (relative to initial suboptimality), we prove the OGM-G method is optimal and uniquely so among fixed-step first-order methods. Finally, for the setting measuring the final squared gradient norm (relative to the initial squared distance to a minimizer), we show the recently proposed Lemniscate method is optimal and uniquely so. Our proofs rely on algebraic reductions for lower bound arguments rather than traditional information-theoretic bounds, which were previously only able to establish OGM's optimality but not uniqueness.
Related
- Sliding Methods for Hölder-Smooth Convex--Concave Minimax Optimization with Bilinear Coupling
- Minimax optimality for sequential gradient-free minimization of smooth functions and their derivatives
- Minimax Optimality of Shampoo for Nonstationary Stochastic Optimization
- Obtaining Game-Stationary Points for Smooth Nonconvex-Nonconcave Minimax Problems via First-Order Methods
- Sharp Minimax Rates for Highly Smooth Strongly Convex Zeroth-Order Optimization
BibTeX
@misc{grimmer2026minimax,
title = {{On Minimax Optimality and Uniqueness of Fixed-Step First-Order Methods for Smooth Convex Optimization}},
author = {Benjamin Grimmer and Sunghyeon Jo and Chanwoo Park},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07671},
eprint = {2609.07671},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07671},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Minimax Optimality and Uniqueness of Fixed-Step First-Order Methods for Smooth Convex Optimization
AU - Benjamin Grimmer
AU - Sunghyeon Jo
AU - Chanwoo Park
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07671
UR - https://arxiv.org/abs/2609.07671
AB - This paper considers the design of optimal fixed-step first-order methods for high-dimensional minimization of L-smooth convex functions. For optimizing worst-case performance measured via suboptimality of the final function value (relative to the initial squared distance to a minimizer), we provide an algebraic proof of the optimality of the optimized gradient method (OGM) and establish its uniqueness among all fixed-step first-order methods. For the alternative measure of final squared gradient norm (relative to initial suboptimality), we prove the OGM-G method is optimal and uniquely so among fixed-step first-order methods. Finally, for the setting measuring the final squared gradient norm (relative to the initial squared distance to a minimizer), we show the recently proposed Lemniscate method is optimal and uniquely so. Our proofs rely on algebraic reductions for lower bound arguments rather than traditional information-theoretic bounds, which were previously only able to establish OGM's optimality but not uniqueness.
ER -