Gradient-Variation Bound for Online Convex Optimization with Constraints

We study online convex optimization with constraints consisting of multiple functional and a relatively simple constraint set, such as Euclidean ball. As enforcing the at each time step through projections is computationally challenging in general, we allow decisions to violate but aim achieve low regret cumulative violation over horizon T steps. First-order methods an O(sqrt{T}) O(1) violation, which best-known bound under Slater's condition, do not take into account structural information problem. Furthermore, existing algorithms analysis are limited space. In this paper, provide instance-dependent for complex obtained by novel primal-dual mirror-prox algorithm. Our quantified total gradient variation V_*(T) sequence loss functions. The proposed algorithm works general normed spaces simultaneously achieves O(sqrt{V_*(T)}) never worse than (O(sqrt{T}), O(1)) result improves previous that applied mirror-prox-type problem achieving O(T^{2/3}) violation. Finally, our efficient, it only performs mirror descent steps iteration instead solving Lagrangian minimization