Optimistic Online Mirror Descent for Bridging Stochastic and Adversarial Online Convex Optimization
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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 σ_1:T^2 and the cumulative adversarial variation Σ_1:T^2 for convex functions. They also provide a slightly weaker bound based on the maximal stochastic variance σ_max^2 and the maximal adversarial variation Σ_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 𝒪(√(σ_1:T^2)+√(Σ_1:T^2)) regret bound, without the convexity requirement of individual functions. For strongly convex and smooth functions, we establish an 𝒪(min{log (σ_1:T^2+Σ_1:T^2), (σ_max^2 + Σ_max^2) log T}) bound, better than their 𝒪((σ_max^2 + Σ_max^2) log T) bound. For and smooth functions, we achieve a new 𝒪(dlog(σ_1:T^2+Σ_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.
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