Optimal Algorithms for Online Convex Optimization with Adversarial Constraints

A well-studied generalization of the standard online convex optimization (OCO) is constrained online convex optimization (COCO). In COCO, on every round, a convex cost function and a convex constraint function are revealed to the learner after the action for that round is chosen. The objective is to design an online policy that simultaneously achieves a small regret while ensuring a small cumulative constraint violation (CCV) against an adaptive adversary interacting over a horizon of length T. A long-standing open question in COCO is whether an online policy can simultaneously achieve O(√(T)) regret and O(√(T)) CCV without any restrictive assumptions. For the first time, we answer this in the affirmative and show that an online policy can simultaneously achieve O(√(T)) regret and Õ(√(T)) CCV. Furthermore, in the case of strongly convex cost and convex constraint functions, the regret guarantee can be improved to O(log T) while keeping the CCV bound the same as above. We establish these results by effectively combining the adaptive regret bound of the AdaGrad algorithm with Lyapunov optimization - a classic tool from control theory. Surprisingly, the analysis is short and elegant.