AI 中文总结
本研究刻画了有限和非凸-强凹优化的极小极大查询复杂度，提出递归近端下降-上升算法，在均方平均光滑性下达到近最优上界，并证明匹配下界。
AI 中文摘要
我们在均方平均光滑性下，刻画了有限和非凸-强凹优化的极小极大期望查询复杂度，直至对数因子。对于随机零尊重增量一阶算法，在精确对偶初始化和$0<\varepsilon^2\le c_0L\Delta$（其中$c_0>0$为普适常数）条件下，复杂度为$\tilde{\Theta}(n+\min\{\sqrt{n}\\,\kappa,n^{3/4}\sqrt{\kappa}\}\\,L\Delta\varepsilon^{-2})$，对所有$\kappa=L/\mu\ge1$成立。这一刻画是我们的主要结果。对偶链构造提供了下界，并在$\kappa\ge\sqrt{n}$时匹配现有Catalyst上界。对于$1\le\kappa\le\sqrt{n}$，我们引入了递归近端下降-上升（RPDA）算法，该算法在近端子问题间保留递归梯度估计器，并以固定查询预算达到$\tilde{O}(n+\sqrt{n}\\,\kappa L\Delta\varepsilon^{-2})$。两个上界均保证期望平方原始梯度至多为$\varepsilon^2$。在个体光滑性下，我们还证明了对于所有$\kappa\ge1$，复杂度为$\Omega(n+\min\{\kappa,\sqrt{n\kappa}\}\\,L\Delta\varepsilon^{-2})$。当$\kappa\ge n$时，该下界匹配双线性子类的多项式上界；中间条件数差距仍然开放。
英文摘要
We characterize, up to logarithmic factors, the minimax expected query complexity of finite-sum nonconvex-strongly-concave optimization under mean-squared averaged smoothness. For randomized zero-respecting incremental first-order algorithms, the complexity is $\tildeΘ(n+\min\{\sqrt{n}\,κ,n^{3/4}\sqrtκ\}\,LΔ\varepsilon^{-2})$ throughout $κ=L/μ\ge1$, under exact dual initialization and for $0<\varepsilon^2\le c_0LΔ$, where $c_0>0$ is universal. This characterization is our main result. A dual-chain construction provides the lower bound and matches the existing Catalyst upper bound for $κ\ge\sqrt{n}$. For $1\leκ\le\sqrt{n}$, we introduce recursive proximal descent-ascent (RPDA), which retains a recursive gradient estimator across proximal subproblems and attains $\tilde{O}(n+\sqrt{n}\,κLΔ\varepsilon^{-2})$ with a fixed query budget. Both upper bounds guarantee an expected squared primal gradient at most $\varepsilon^2$. Under individual smoothness, we also prove $Ω(n+\min\{κ,\sqrt{nκ}\}\,LΔ\varepsilon^{-2})$ for every $κ\ge1$. It matches the polynomial upper rate for a bilinear subclass when $κ\ge n$; the intermediate-condition-number gap remains open.
Comments30 pages, 2 figures, 2 tables
