发表机构
Cornell University（康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对非光滑随机凸-凹鞍点问题，提出模块化单循环无投影算法，利用线性最小化预言机直接处理非光滑性，达到O(ε^-2)迭代复杂度并证明极小极大最优性。
AI 中文摘要
我们研究紧凸集上的非光滑凸-凹鞍点问题，假设可以访问支付函数的随机次梯度。我们开发了使用原始域和对偶域上的线性最小化预言机的单循环无投影算法。与先前依赖平滑化的无投影方法不同，我们的方法纯粹基于次梯度，直接处理非光滑性。这种设计使框架具有模块化性：当在特定域上不使用线性最小化预言机时，相应的更新可以替换为标准投影预言机，而无需改变单循环结构。因此，我们的框架涵盖了完全无投影和混合预言机配置。我们在标准无偏随机预言机假设（具有有界方差）下证明了任意时间的强鞍间隙保证。我们的算法实现了$O(\epsilon^{-2})$的迭代复杂度，与投影随机次梯度方法相匹配。我们还为相应的预言机模型提供了匹配的下界，确立了极小极大最优性。我们的结果表明，对于非光滑随机极小极大问题，线性最小化预言机的计算优势不必以统计或预言机效率为代价。
英文摘要
We study nonsmooth convex-concave saddle-point problems over compact convex sets, assuming access to stochastic subgradients of the payoff function. We develop single-loop projection-free algorithms that use linear minimization oracles over the primal and dual domains. Unlike prior projection-free approaches that rely on smoothing, our methods are purely subgradient-based and handle nonsmoothness directly. This design makes the framework modular: when a linear minimization oracle is not used over a specific domain, the corresponding update can be replaced by a standard projection oracle without changing the single-loop structure. Thus, our framework covers both fully projection-free and hybrid oracle configurations. We prove anytime strong saddle gap guarantees under standard unbiased stochastic oracle assumptions with bounded variance. Our algorithms achieve an $O(ε^{-2})$ iteration complexity, matching projected stochastic subgradient methods. We also provide matching lower bounds for the corresponding oracle models, establishing minimax optimality. Our results show that, for nonsmooth stochastic minimax problems, the computational advantages of linear minimization oracles need not come at the expense of statistical or oracle efficiency.
