Stochastically Extended Adversarial (SEA) model is introduced by Sachs et al. [2022] as an interpolation between stochastic and adversarial online convex optimization. Under the smoothness condition, they demonstrate that the expected regret of optimistic follow-the-regularized-leader (FTRL) depends on the cumulative stochastic variance $\sigma_{1:T}^2$ and the cumulative adversarial variation $\Sigma_{1:T}^2$ for convex functions. They also provide a slightly weaker bound based on the maximal stochastic variance $\sigma_{\max}^2$ and the maximal adversarial variation $\Sigma_{\max}^2$ for strongly convex functions. Inspired by their work, we investigate the theoretical guarantees of optimistic online mirror descent (OMD) for the SEA model. 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 convex and smooth functions, we establish an $\mathcal{O}((\sigma_{\max}^2 + \Sigma_{\max}^2) \log (\sigma_{1:T}^2+\Sigma_{1:T}^2))$ bound, better than their $\mathcal{O}((\sigma_{\max}^2 + \Sigma_{\max}^2) \log T)$ result. For exp-concave and smooth functions, we achieve a new $\mathcal{O}(d\log(\sigma_{1:T}^2+\Sigma_{1:T}^2))$ bound. Owing to the OMD framework, we broaden our work to study dynamic regret minimization and scenarios where the online functions are non-smooth. We establish the first dynamic 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, we propose novel algorithms building on optimistic OMD with an implicit update, which provably attain static regret and dynamic regret guarantees without smoothness conditions.

翻译：Sachs等人[2022]提出的随机扩展对抗（SEA）模型作为随机与对抗在线凸优化之间的插值模型。在光滑性条件下，他们证明了对于凸函数，乐观追随正则化领导者（FTRL）的期望遗憾取决于累积随机方差$\sigma_{1:T}^2$和累积对抗变差$\Sigma_{1:T}^2$。他们还针对强凸函数给出了一个基于最大随机方差$\sigma_{\max}^2$和最大对抗变差$\Sigma_{\max}^2$的稍弱界。受其启发，我们研究了乐观在线镜像下降（OMD）在SEA模型中的理论保证。对于凸且光滑的函数，我们在不需要单个函数凸性要求的情况下，获得了相同的$\mathcal{O}(\sqrt{\sigma_{1:T}^2}+\sqrt{\Sigma_{1:T}^2})$遗憾界。对于强凸且光滑的函数，我们建立了$\mathcal{O}((\sigma_{\max}^2 + \Sigma_{\max}^2) \log (\sigma_{1:T}^2+\Sigma_{1:T}^2))$的界，优于他们的$\mathcal{O}((\sigma_{\max}^2 + \Sigma_{\max}^2) \log T)$结果。对于指数凹且光滑的函数，我们实现了新的$\mathcal{O}(d\log(\sigma_{1:T}^2+\Sigma_{1:T}^2))$界。借助OMD框架，我们将工作扩展到动态遗憾最小化以及在线函数非光滑的场景。我们首次为具有凸且光滑函数的SEA模型建立了动态遗憾保证，在非平稳场景下优于静态遗憾界。此外，为了处理SEA模型中的非光滑凸函数，我们基于带有隐式更新的乐观OMD提出了新算法，该算法在无光滑性条件下可证明地实现静态和动态遗憾保证。