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

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(2)(1:T) and the cumulative adversarial variation Sigma(2)(1:T) for convex functions. They also provide a slightly weaker bound based on the maximal stochastic variance sigma(2)(max) and the maximal adversarial variation Sigma(2)(max) max 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 O(root sigma(2)(1:T) + root Sigma(2)(1:T)) regret bound, without the convexity requirement of individual functions. For strongly convex and smooth functions, we establish an O(min{log(sigma(2)(1:T) + Sigma(2)(1:T)), (sigma(2)(max) + Sigma(2)(max)) log T}) bound, better than their O((sigma(2)(max) + Sigma(2)(max)) log T) result. For exp-concave and smooth functions, we achieve a new O(d log(sigma(2)(1:T) + Sigma(2)(1:T))) bound. Owing to the OMD framework, we further establish dynamic regret for convex and smooth functions, which is more favorable in non-stationary online scenarios.