A Unified Analysis Method for Online Optimization in Normed Vector Space
This work addresses the need for a more general and tighter analysis in online optimization, which is incremental as it builds upon and improves existing methods.
The paper tackles the problem of online optimization by providing a unified theoretical analysis that generalizes existing algorithms and tightens regret bounds, achieving better results than known optimal ones for specific instantiations like normalized exponentiated subgradient and greedy/lazy projection.
This paper studies online optimization from a high-level unified theoretical perspective. We not only generalize both Optimistic-DA and Optimistic-MD in normed vector space, but also unify their analysis methods for dynamic regret. Regret bounds are the tightest possible due to the introduction of $φ$-convex. 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.