AI 中文总结
研究给定条件下 Lipschitz 凸优化的最优固定步长一阶方法，给出其完整刻画，证明可从构造法导出，还通过证明乘数给出最优方法集的多面体表示，并表明不存在随时最优固定步长子梯度方法。
AI 中文摘要
我们考虑给定\(\|x_0 - x_*\|\leq D\)时，用于\(M -\)Lipschitz 凸优化的最优固定步长一阶方法的设计。先前工作已识别出几种不同的固定步长方法，由步长矩阵\(W\)参数化，目标差距收敛的（信息论）极小极大最优速率为\(MD / \sqrt{N + 1}\)。我们给出了每种最优固定步长方法的完整刻画。此外，表明每种最优固定步长方法都可从[构造性方法]的构造方法导出，并通过证明乘数给出最优方法集的多面体表示。由此刻画可知不存在随时最优固定步长子梯度方法。
英文摘要
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.
Comments10 pages
