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Stochastically Extended Adversarial (SEA) model is introduced by Sachs et al.[2022] as an interpolation between stochastic and adversarial online convexoptimization. Under the smoothness condition, they demonstrate that theexpected regret of optimistic follow-the-regularized-leader (FTRL) depends onthe cumulative stochastic variance $\sigma_{1:T}^2$ and the cumulativeadversarial variation $\Sigma_{1:T}^2$ for convex functions. They also providea slightly weaker bound based on the maximal stochastic variance$\sigma_{\max}^2$ and the maximal adversarial variation $\Sigma_{\max}^2$ forstrongly convex functions. Inspired by their work, we investigate thetheoretical guarantees of optimistic online mirror descent (OMD) for the SEAmodel. For convex and smooth functions, we obtain the same$\mathcal{O}(\sqrt{\sigma_{1:T}^2}+\sqrt{\Sigma_{1:T}^2})$ regret bound,without the convexity requirement of individual functions. For strongly convexand smooth functions, we establish an $\mathcal{O}((\sigma_{\max}^2 +\Sigma_{\max}^2) \log (\sigma_{1:T}^2+\Sigma_{1:T}^2))$ bound, better thantheir $\mathcal{O}((\sigma_{\max}^2 + \Sigma_{\max}^2) \log T)$ result. Forexp-concave and smooth functions, we achieve a new$\mathcal{O}(d\log(\sigma_{1:T}^2+\Sigma_{1:T}^2))$ bound. Owing to the OMDframework, we broaden our work to study dynamic regret minimization andscenarios where the online functions are non-smooth. We establish the firstdynamic regret guarantee for the SEA model with convex and smooth functions,which is more favorable than static regret bounds in non-stationary scenarios.Furthermore, to deal with non-smooth and convex functions in the SEA model, wepropose novel algorithms building on optimistic OMD with an implicit update,which provably attain static regret and dynamic regret guarantees withoutsmoothness conditions.

Optimistic Online Mirror Descent for Bridging Stochastic and Adversarial Online Convex Optimization