  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.
