AI 中文总结
该研究针对非凸-PŁ极小极大优化，证明确定性一阶方法寻找满足梯度范数不超过ε的平稳点时，预言机查询下界为Ω(ℓΔκ/ε²)，且该下界与已知上界匹配，确定了κ的线性依赖不可避免。
AI 中文摘要
我们研究光滑非凸-Polyak-Łojasiewicz（NC-PŁ）极小极大优化中，寻找值函数平稳点的确定性一阶预言机复杂度。假设目标函数是联合ℓ-光滑的，且在对偶变量上满足μ-PŁ条件，其值函数Φ(x):=max_y f(x;y)满足Φ(0)-inf_x Φ(x)≤Δ。当κ:=ℓ/μ≳1且0<ε²≲ℓΔ时，我们证明，最坏情况下，每个确定性一阶方法需要Ω(ℓΔκ/ε²)次预言机查询，才能找到满足||∇Φ(x)||≤ε的x。该速率在对(ℓ,Δ,κ,ε)的依赖上与已知上界[Yang等人，2022]匹配，表明确定性一阶方法对κ的线性依赖是不可避免的。
英文摘要
We study the deterministic first-order oracle complexity of finding stationary points of the value function in smooth nonconvex-Polyak-Łojasiewicz (NC-PŁ) minimax optimization. We assume that the objective is jointly $\ell$-smooth and satisfies the $μ$-PŁ condition in the dual variable, and that its value function $Φ(x):=\max_y f(x;y)$ satisfies $Φ(0)-\inf_xΦ(x)\leqΔ$. When $κ:=\ell/μ\gtrsim 1$ and $0<ε^2\lesssim\ellΔ$, we prove that every deterministic first-order method requires $Ω(\ellΔκ/ε^2)$ oracle queries in the worst case to find $x$ satisfying $\|\nablaΦ(x)\|\leqε$. This rate matches the known upper bound in its dependence on $(\ell,Δ,κ,ε)$ [Yang et al., 2022] and shows that the linear dependence on $κ$ is unavoidable for deterministic first-order methods.
