On Minimax Optimality and Uniqueness of Fixed-Step First-Order Methods for Smooth Convex Optimization

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.