AI 中文总结
本文证明非凸-强凹极小极大优化中SAPD+的随机复杂度对条件数呈线性依赖，并给出匹配下界，揭示该依赖的必然性。
AI 中文摘要
我们研究SAPD+在非凸-强凹极小极大优化中的随机复杂度是否必然依赖线性条件数。对于联合$L$-光滑且具有对偶强凹参数$\mu$的目标函数，我们在相同的Moreau包络平稳性准则和相同的原始-对偶初始化间隙下，证明了一个与SAPD+上界匹配的下界。具体而言，当$\sigma\ge\varepsilon$时，在所述精度范围内，零尊重算法的最坏情况复杂度为$\Theta(\kappa LG\sigma^2\varepsilon^{-4})$，其中$\kappa=L/\mu$，$G$限制初始原始-对偶间隙，$\sigma^2$限制一般无偏一阶预言机的方差。该下界在一个具有有界对偶盒的光滑问题类上实现。我们的构造将非凸零链的每个链接通过对偶梯度路由，其大小与$\varepsilon/\sqrt{\kappa}$成正比，而未发现的原始坐标阻止平稳性。结合已知的确定性下界，它还产生原始梯度下界$\Omega(L\Delta(\sqrt{\kappa}\varepsilon^{-2}+\kappa\sigma^2\varepsilon^{-4}))$，其中$\Delta$限制初始原始函数间隙。
英文摘要
We study whether the linear condition-number dependence in the stochastic complexity of SAPD+ is necessary for nonconvex-strongly-concave minimax optimization. For jointly $L$-smooth objectives with dual strong-concavity parameter $μ$, we prove a lower bound that matches the SAPD+ upper bound under the same Moreau-envelope stationarity criterion and the same primal-dual initialization gap. Specifically, when $σ\ge\varepsilon$, the worst-case complexity of zero-respecting algorithms is $Θ(κLGσ^2\varepsilon^{-4})$ in the stated accuracy regime, where $κ=L/μ$, $G$ bounds the initial primal-dual gap, and $σ^2$ bounds the variance of a general unbiased first-order oracle. The lower bound is realized on a smooth problem class with a bounded dual box. Our construction routes each link of a nonconvex zero-chain through a dual gradient of magnitude proportional to $\varepsilon/\sqrtκ$, while an undiscovered primal coordinate prevents stationarity. It also yields the primal-gradient lower bound $Ω(LΔ(\sqrtκ\varepsilon^{-2}+κσ^2\varepsilon^{-4}))$ after combination with the known deterministic bound, where $Δ$ bounds the initial primal function gap.
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