Online Convex Optimization for Cumulative Constraints

We propose an algorithm for online convex optimization which examines a clipped long-term constraint of the form T ∑ t=1 [g(xt)]+, encoding the cumulative constraint violation. Previous literature has focused on longterm constraints of the form T ∑ t=1 g(xt), for which strictly feasible solutions can cancel out the effects of violated constraints. In this paper, we generalize the results in [16] and [14] by showing that the clipped one T ∑ t=1 [g(xt)]+ also has the same upper bound as the average one T ∑ t=1 g(xt). The performance of our proposed algorithm is also tested in experiments, in which we find that our algorithm can either follow the boundary of the constraints tightly or have relatively low clipped constraint violation.