AI 中文总结
该研究确立了弱凸优化的最优确定性神谕复杂度，揭示了光滑非凸与非光滑弱凸优化间的复杂度差异，为相关优化问题的算法设计提供了理论依据。
AI 中文摘要
我们研究ρ-弱凸且G-Lipschitz函数的ε-驻点查找的神谕复杂度，其中驻点通过Moreau包络的梯度衡量。我们考虑一阶神谕，其在每个查询点返回函数值和全次梯度。我们证明，当Δ≤G²/ρ（其中f(z)-inf f≤Δ）时，每个确定性一阶算法需要Ω(ρG²Δ/ε⁴)次神谕查询。该下界在通用常数范围内与已知最优确定性和随机一阶上界匹配，确立了最优确定性神谕复杂度。该结果揭示了光滑非凸与非光滑弱凸优化之间的基本复杂度差异：光滑非凸最小化的神谕复杂度为Θ(ε⁻²)，而非光滑弱凸优化因非光滑几何而非随机性，会产生额外的内在ε⁻²因子。
英文摘要
We study the oracle complexity of finding $ε$-stationary points of $ρ$-weakly convex and $G$-Lipschitz functions, where stationarity is measured by the gradient of the Moreau envelope. We consider a first-order oracle that returns both the function value and the full subdifferential at every query point. We prove that every deterministic first-order algorithm requires $ Ω({ρG^2Δ}/{ε^4})$ oracle queries whenever $Δ\leq {G^2}/ρ$, where $f(\bz)-\inf f \leq Δ$. This lower bound matches the best known deterministic and stochastic first-order upper bounds, up to universal constants, and establishes the optimal deterministic oracle complexity. The result reveals a fundamental complexity separation between smooth nonconvex and nonsmooth weakly convex optimization. While smooth nonconvex minimization admits a $Θ(ε^{-2})$ oracle complexity, nonsmooth weakly convex optimization incurs an intrinsic additional $ε^{-2}$ factor arising from nonsmooth geometry rather than stochasticity.
