发表机构
Technion - Israel Institute of Technology（以色列理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究基于弱近端预言机的复合凸优化方法，证明其无法加速，并给出新的上界，指出其预言机复杂性存在固有损失。
AI 中文摘要
我们考虑一个标准的凸复合优化问题，其目标函数可以是光滑的或非光滑的，并满足二次增长条件。近年来，多项工作提出了基于弱近端预言机（WPO）的算法，这些算法在预言机复杂性上基本匹配了依赖精确近端操作的近端（次）梯度方法。重要的是，这种WPO放宽了标准近端算子的强最优性条件，当最优解具有某种稀疏结构时，其运行时实现可能更加高效。一个遗留问题是，这种基于WPO的方法能否（在Nesterov加速梯度的意义上）被加速。在本工作中，我们通过建立针对确定性和随机性方法的下界，给出了否定答案。因此，虽然WPO可以显著降低单次预言机调用的成本，但这伴随着预言机复杂性上的固有损失。我们还为基于WPO的非光滑凸复合优化提供了新的上界，该上界几乎匹配近端次梯度方法。
英文摘要
We consider a standard convex composite optimization problem with either smooth or nonsmooth objective function, and under quadratic growth. In recent years, several works gave algorithms based on a \textit{weak proximal oracle} (WPO) that essentially match in oracle complexities proximal (sub)gradient methods relying on exact prox operations. Importantly, such WPOs, which relax the strong optimality condition of the standard prox operator, may admit much more efficient implementation in terms of runtime when optimal solutions have some sparse structure. A question remained if such WPO-based methods can be accelerated (in the sense of Nesterov's accelerated gradient). In this work we provide a negative answer by establishing lower bounds against both deterministic and randomized methods. Thus, while WPOs can substantially reduce the cost of individual oracle calls, this comes with an inherent loss in oracle complexity. We also provide a new upper-bound for WPO-based nonsmooth convex composite optimization, nearly matching the proximal subgradient method.
