欧几里得球上零阶非凸优化的联合下界

中文摘要 AI 辅助

我们证明了在欧几里得查询球上，即使保证球内包含一个驻点，Goldstein 平稳性的联合随机零阶下界仍然成立。在维度 $d$ 中，设 $f=\mathbb{E}[F(\cdot;\xi)]$，假设 $\mathbb{E}[\operatorname{Lip}(F(\cdot;\xi))^2]\le L_0^2$，并将球上的初始目标差距限制为 $\Delta$。对于邻域半径 $\delta>0$ 和残差容限 $\varepsilon>0$，我们的光滑困难族要求 $\Omega(dL_0^2\Delta/(\delta\varepsilon^3))$ 次标量评估，成功概率为 $1/2$，针对可能保留并重复查询每个采样函数的随机自适应算法。该结果在构造的半径为 $\Theta(\Delta/\varepsilon)$ 的球上成立，条件为 $\varepsilon\le cL_0$，$\Delta\ge C\delta\varepsilon$，且 $d\ge C[1+\log(2+\Delta L_0^2/(\delta\varepsilon^3))]$。一系列局部化区域迫使重复的方向估计，自适应高斯后验论证控制样本重用。该结果建立了可解有界域实例上的联合维度和精度障碍。该差距局限于查询球；在公共半径处的匹配极小极大刻画以及相应的无限制全局差距下界在此仍然开放。

英文摘要

We prove a joint stochastic zeroth-order lower bound for Goldstein stationarity on a Euclidean query ball, even when the ball is guaranteed to contain a stationary point. In dimension $d$, let $f=\mathbb{E}[F(\cdot;ξ)]$, assume $\mathbb{E}[\operatorname{Lip}(F(\cdot;ξ))^2]\le L_0^2$, and bound the initial objective gap over the ball by $Δ$. For neighborhood radius $δ>0$ and residual tolerance $\varepsilon>0$, our smooth hard family requires $Ω(dL_0^2Δ/(δ\varepsilon^3))$ scalar evaluations for success probability $1/2$, against randomized adaptive algorithms that may retain and repeatedly query each sampled function. The result holds for $\varepsilon\le cL_0$, $Δ\ge Cδ\varepsilon$, and $d\ge C[1+\log(2+ΔL_0^2/(δ\varepsilon^3))]$, on a constructed ball of radius $Θ(Δ/\varepsilon)$. A sequence of localized regions forces repeated direction estimation, and an adaptive Gaussian posterior argument controls sample reuse. The result establishes a joint dimension and accuracy obstruction on solvable bounded-domain instances. The gap is local to the query ball; a matching minimax characterization at a common radius and the corresponding unrestricted global-gap lower bound remain open here.

发表机构

• Peking University（北京大学）

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