A Unified Analysis Method for Online Optimization in Normed Vector Space

We present a unified analysis method that relies on the generalized cosine rule and $\phi$-convex for online optimization in normed vector space using dynamic regret as the performance metric. In combing the update rules, we start with strategy $S$ (a two-parameter variant strategy covering Optimistic-FTRL with surrogate linearized losses), and obtain $S$-I (type-I relaxation variant form of $S$) and $S$-II (type-II relaxation variant form of $S$, which is Optimistic-MD) by relaxation. Regret bounds for $S$-I and $S$-II are the tightest possible. As instantiations, regret bounds of normalized exponentiated subgradient and greedy/lazy projection are better than the currently known optimal results. By replacing losses of online game with monotone operators, and extending the definition of regret, namely regret$^n$, we extend online convex optimization to online monotone optimization, which expands the application scope of $S$-I and $S$-II.