Centered around solving the Online Saddle Point problem, this paper introduces the Online Convex-Concave Optimization (OCCO) framework, which involves a sequence of two-player time-varying convex-concave games. We propose the generalized duality gap (Dual-Gap) as the performance metric and establish the parallel relationship between OCCO with Dual-Gap and Online Convex Optimization (OCO) with regret. To demonstrate the natural extension of OCCO from OCO, we develop two algorithms, the implicit online mirror descent-ascent and its optimistic variant. Analysis reveals that their duality gaps share similar expression forms with the corresponding dynamic regrets arising from implicit updates in OCO. Empirical results further substantiate the effectiveness of our algorithms. Simultaneously, we unveil that the dynamic Nash equilibrium regret, which was initially introduced in a recent paper, has inherent defects.

翻译：围绕求解在线鞍点问题，本文引入了在线凸凹优化（OCCO）框架，该框架涉及一系列双人时变凸凹博弈。我们提出广义对偶间隙（Dual-Gap）作为性能度量，并建立了基于对偶间隙的OCCO与基于遗憾的在线凸优化（OCO）之间的并行关系。为展示OCCO是OCO的自然延伸，我们开发了两种算法：隐式在线镜像下降-上升算法及其乐观变体。分析表明，它们的对偶间隙与OCO中隐式更新产生的相应动态遗憾具有相似的表达形式。实证结果进一步验证了我们算法的有效性。同时，我们揭示了近期论文中首次提出的动态纳什均衡遗憾存在固有缺陷。