Goldstein 稳定性的指数确定性查询复杂度
Exponential Deterministic Query Complexity of Goldstein Stationarity
- Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
- University of Chinese Academy of Sciences（中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究非光滑非凸 Lipschitz 函数的 Goldstein 稳定点查询复杂度，证明在固定小精度下维度指数下界，并给出相应上界及无维度算法。
AI中文摘要：
我们研究了寻找全局 Lipschitz 函数（可能非光滑且非凸）的近似 Goldstein 稳定点的确定性查询复杂度。在 Lipschitz 常数和初始函数值与下确界之差的给定界限下，我们证明了在固定足够小的精度参数下，最坏情况下 oracle 调用次数的下界随维度呈指数增长。该结果适用于局部 oracle，并给出了对精度参数的显式依赖。一个仅使用函数值的互补确定性算法给出了指数上界，从而确立了该机制下维度上的指数阶。对于较粗略的稳定性要求，我们还给出了一种具有无维度查询界限的确定性一阶算法。因此，维度依赖性在两个精度机制之间有所不同。
英文摘要：
We study the deterministic query complexity of finding approximate Goldstein stationary points of globally Lipschitz functions that may be nonsmooth and nonconvex. Under prescribed bounds on the Lipschitz constant and on the difference between the initial function value and the infimum, we prove a lower bound on the worst-case number of oracle calls that is exponential in the dimension at fixed sufficiently small accuracy parameters. The result holds for a local oracle and gives explicit dependence on the accuracy parameters. A complementary deterministic algorithm using only function values gives an exponential upper bound, establishing the exponential order in the dimension in this regime. For a coarser stationarity requirement, we also give a deterministic first-order algorithm with a dimension-free query bound. Thus, the dimension dependence differs between the two accuracy regimes.
