发表机构
Google Research（谷歌研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了加权初始条件下L-光滑强凸函数N次一阶调用后函数值次优性的精确极小极大风险，构造了匹配下界的常数内存方法ITEM-w，并证明了ITEM-f的最优性。
AI 中文摘要
我们确定了在非负加权组合的初始平方距离、函数值次优性和平方梯度范数下，对$L$-光滑、$\mu$-强凸函数进行$N$次一阶预言机调用后的精确最坏情况函数值次优性。排除无法保证函数值次优性严格改进的情况，在维度$d\geq2N+1$下，精确的确定性极小极大风险由涉及$N$步递推的单个标量方程的唯一解刻画。证明为下界构造了一个显式的困难实例，并构造了一种常数内存的一阶方法ITEM-w，其最坏情况性能与该下界匹配。在标准初始条件下，该结果恢复了Kim和Fessler的优化梯度方法所达到的精确凸界，并确立了Kim、Ryu和Das Gupta最近提出的ITEM-f方法在初始函数值条件下的极小极大最优性。
英文摘要
We determine the exact worst-case function-value suboptimality after $N$ first-order oracle calls on $L$-smooth, $μ$-strongly convex functions under a nonnegative weighted combination of the initial squared distance, function-value suboptimality, and squared gradient norm. Excluding the case where no strict improvement in function-value suboptimality can be guaranteed, the exact deterministic minimax risk in dimension $d\geq2N+1$ is characterized by the unique solution of a single scalar equation involving an $N$-step recurrence. The proof constructs an explicit hard instance for the lower bound and a constant-memory first-order method, ITEM-w, whose worst-case performance matches this lower bound. On the standard initial conditions, the result recovers the exact convex bound attained by the Optimized Gradient Method of Kim and Fessler, and establishes the minimax optimality of the recently introduced ITEM-f method of Kim, Ryu, and Das Gupta for the initial function-value condition.
