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 $\mathcal{O}(\min\{\log (\sigma_{1:T}^2+\Sigma_{1:T}^2), (\sigma_{\max}^2 + \Sigma_{\max}^2) \log T\})$ bound, better than their $\mathcal{O}((\sigma_{\max}^2 + \Sigma_{\max}^2) \log T)$ bound. For \mbox{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 can further extend our result to obtain dynamic regret guarantees, which are more favorable in non-stationary online scenarios. The attained results allow us to recover excess risk bounds of the stochastic setting and regret bounds of the adversarial setting, and derive new guarantees for many intermediate scenarios.

翻译：Sachs等人[2022]提出的随机扩展对抗(SEA)模型是随机在线凸优化与对抗在线凸优化之间的插值模型。在光滑性条件下，他们证明乐观后正则化领导(FTRL)方法在凸函数上的期望遗憾依赖于累积随机方差$\sigma_{1:T}^2$和累积对抗变异$\Sigma_{1:T}^2$。对于强凸函数，他们基于最大随机方差$\sigma_{\max}^2$和最大对抗变异$\Sigma_{\max}^2$给出了一个稍弱的上界。受其工作启发，我们研究了乐观在线镜像下降(OMD)方法在SEA模型上的理论保证。对于凸光滑函数，我们在不要求单个函数凸性的条件下得到了相同的$\mathcal{O}(\sqrt{\sigma_{1:T}^2}+\sqrt{\Sigma_{1:T}^2})$遗憾上界。对于强凸光滑函数，我们建立了$\mathcal{O}(\min\{\log (\sigma_{1:T}^2+\Sigma_{1:T}^2), (\sigma_{\max}^2 + \Sigma_{\max}^2) \log T\})$上界，优于其$\mathcal{O}((\sigma_{\max}^2 + \Sigma_{\max}^2) \log T)$上界。对于指数凹光滑函数，我们获得了新的$\mathcal{O}(d\log(\sigma_{1:T}^2+\Sigma_{1:T}^2))$上界。得益于OMD框架，我们进一步将结果推广至动态遗憾保证，这在非平稳在线场景中更具优势。所得结果能够恢复随机设置下的超额风险上界和对抗设置下的遗憾上界，并为众多中间场景推导出新的理论保证。