随机非凸-强凹极小极大优化的紧下界
Tight Lower Bounds for Stochastic Nonconvex-Strongly-Concave Minimax Optimization
- Nanjing University（南京大学）
- the Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对光滑非凸-强凹极小极大优化，建立了随机一阶预言机复杂度的紧下界，改进了条件数依赖，并首次给出平均光滑性下界，证明基于二次提升构造。
AI中文摘要：
我们研究了在光滑非凸-强凹极小极大优化中，寻找原始函数的ε-驻点的随机一阶预言机复杂度。对于足够小的ε，在有界方差假设下，我们建立了Ω(κLΔσ²ε⁻⁴)的下界；在额外假设平均光滑性下，建立了Ω(κ^{3/2}L̄Δσε⁻³)的下界。这里，L和L̄分别表示光滑常数和平均光滑常数，Δ是初始原始间隙，σ²限制预言机方差，κ=L/μ或L̄/μ分别对应相应设置，其中μ是强凹参数。我们的有界方差下界将条件数依赖从先前下界的κ^{1/3}改进为κ，而我们的平均光滑性下界是同类中的第一个。在两种设置中，所得下界在κ和ε的依赖上与现有上界匹配。我们的证明基于一个统一的二次提升构造，该构造将随机非凸最小化的困难实例转移到无约束极小极大优化，同时保持所需的方差和光滑性性质。
英文摘要：
We study the stochastic first-order oracle complexity of finding $ε$-stationary points of the primal function in smooth nonconvex-strongly-concave minimax optimization. For sufficiently small $ε$, we establish lower bounds of $Ω(κLΔσ^2ε^{-4})$ under the bounded-variance assumption and $Ω(κ^{3/2}\bar LΔσε^{-3})$ under the additional assumption of averaged smoothness. Here, $L$ and $\bar L$ denote the smoothness and averaged-smoothness constants, respectively, $Δ$ is the initial primal gap, $σ^2$ bounds the oracle variance, and $κ=L/μ$ or $\bar L/μ$ in the respective settings, where $μ$ is the strong-concavity parameter. Our bounded-variance lower bound improves the dependence on the condition number from $κ^{1/3}$ in previous lower bounds to $κ$, while our averaged-smoothness lower bound is the first of its kind. In both settings, the resulting lower bounds match existing upper bounds in their dependence on $κ$ and $ε$. Our proofs are based on a unified quadratic lifting construction that transfers a hardness instance for stochastic nonconvex minimization to unconstrained minimax optimization while preserving the required variance and smoothness properties.
