← 研踪 PaperTrailarXiv:2610.19652 · v2
On Global Joint Lower Bounds in Zeroth-Order Nonconvex Optimization
首次提交 2026-10-14 14:20
版本更新 2026-11-03 23:38
雷达观测 2026-11-04 23:39
摘要
Must the dimension and accuracy costs of zeroth-order optimization multiply? Global upper bounds for smooth stochastic nonconvex optimization scale as $O(dε^{-4})$, with both factors separately necessary. We show that their product is unavoidable. We establish an $Ω(dε^{-4})$ lower bound for smooth stochastic nonconvex optimization, yielding an $Ω(dδ^{-1}ε^{-3})$ bound for nonsmooth Goldstein stationarity. Both match the best-known full-space upper bounds, establishing joint minimax optimality in dimension and accuracy. These bounds hold against fully adaptive randomized algorithms with unrestricted queries and outputs, even under arbitrary sample reuse. As a byproduct, we establish noiseless exact-value lower bounds for both smooth and Goldstein stationarity, even against randomized algorithms.
观测到的版本
相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。
v2 · 2026-11-03 23:38 · 最新观测版本
On Global Joint Lower Bounds in Zeroth-Order Nonconvex Optimization
Must the dimension and accuracy costs of zeroth-order optimization multiply? Global upper bounds for smooth stochastic nonconvex optimization scale as $O(dε^{-4})$, with both factors separately necessary. We show that their product is unavoidable. We establish an $Ω(dε^{-4})$ lower bound for smooth stochastic nonconvex optimization, yielding an $Ω(dδ^{-1}ε^{-3})$ bound for nonsmooth Goldstein stationarity. Both match the best-known full-space upper bounds, establishing joint minimax optimality in dimension and accuracy. These bounds hold against fully adaptive randomized algorithms with unrestricted queries and outputs, even under arbitrary sample reuse. As a byproduct, we establish noiseless exact-value lower bounds for both smooth and Goldstein stationarity, even against randomized algorithms.
查看此版本解读 →查看此版本原文 ↗