提出在线极小极大优化新框架，解决传统均衡与个体遗憾不兼容问题。
Online Min-Max Optimization: From Individual Regrets to Cumulative Saddle Points
- 基于在线凸优化思路，设计静态对偶间隙与动态鞍点遗憾指标
- 在强凸强凹及极小极大指数凹性条件下实现可证明的性能界
- 适用于投资组合等双人博弈场景，支持个体遗憾兼容的动态分析
本文研究一种基于累积鞍点的在线极小极大优化框架，超越经典凸-凹设定。首先指出即使在强凸-强凹函数下，传统静态纳什均衡遗憾（SNE-Reg$_T$）仍与个体遗憾不兼容。为此提出受在线凸优化启发的静态对偶间隙（SDual-Gap$_T$）作为替代指标。通过将问题转化为经典在线凸优化，设计算法实现了对SDual-Gap$_T$和新型动态鞍点遗憾（DSP-Reg$_T$）的上界。后者被建议为在线凸优化中动态遗憾的极小极大版本。在强凸-强凹及极小极大指数凹性（min-max EC）条件下建立性能边界，并揭示一类满足min-max EC的函数类，涵盖经典的双人投资组合选择问题变体。最后，在双侧Polyak-Łojasiewicz（PL）条件下，为与个体遗憾兼容的动态遗憾提供边界。
原文摘要 · Abstract (English)
We propose and study an online version of min-max optimization based on cumulative saddle points under a variety of performance measures beyond convex-concave settings. After first observing the incompatibility of (static) Nash equilibrium (SNE-Reg$_T$) with individual regrets even for strongly convex-strongly concave functions, we propose an alternate \emph{static} duality gap (SDual-Gap$_T$) inspired by the online convex optimization (OCO) framework. We provide algorithms that, using a reduction to classic OCO problems, achieve bounds for SDual-Gap$_T$~and a novel \emph{dynamic} saddle point regret (DSP-Reg$_T$), which we suggest naturally represents a min-max version of the dynamic regret in OCO. We derive our bounds for SDual-Gap$_T$~and DSP-Reg$_T$~under strong convexity-strong concavity and a min-max notion of exponential concavity (min-max EC), and in addition we establish a class of functions satisfying min-max EC~that captures a two-player variant of the classic portfolio selection problem. Finally, for a dynamic notion of regret compatible with individual regrets, we derive bounds under a two-sided Polyak-Łojasiewicz (PL) condition.
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