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This paper studies online optimization from a high-level unified theoreticalperspective. We not only generalize both Optimistic-DA and Optimistic-MD innormed vector space, but also unify their analysis methods for dynamic regret.Regret bounds are the tightest possible due to the introduction of$\phi$-convex. As instantiations, regret bounds of normalized exponentiatedsubgradient and greedy/lazy projection are better than the currently knownoptimal results. By replacing losses of online game with monotone operators,and extending the definition of regret, namely regret$^n$, we extend onlineconvex optimization to online monotone optimization.

A Unified Analysis Method for Online Optimization in Normed Vector Space