Regret and Cumulative Constraint Violation Analysis for Online Convex Optimization with Long Term Constraints

This paper considers online convex optimization with long term constraints, where constraints can be violated in intermediate rounds, but need to be satisfied in the long run. The cumulative constraint violation is used as the metric to measure constraint violations, which excludes the situation that strictly feasible constraints can compensate the effects of violated constraints. A novel algorithm is first proposed and it achieves an $\mathcal{O}(T^{\max\{c,1-c\}})$ bound for static regret and an $\mathcal{O}(T^{(1-c)/2})$ bound for cumulative constraint violation, where $c\in(0,1)$ is a user-defined trade-off parameter, and thus has improved performance compared with existing results. Both static regret and cumulative constraint violation bounds are reduced to $\mathcal{O}(\log(T))$ when the loss functions are strongly convex, which also improves existing results. %In order to bound the regret with respect to any comparator sequence, In order to achieve the optimal regret with respect to any comparator sequence, another algorithm is then proposed and it achieves the optimal $\mathcal{O}(\sqrt{T(1+P_T)})$ regret and an $\mathcal{O}(\sqrt{T})$ cumulative constraint violation, where $P_T$ is the path-length of the comparator sequence. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.