Beyond O(T)Constraint Violation for Online Convex Optimization with Adversarial Constraints

We study Online Convex Optimization with adversarial constraints (COCO). At each round a learner selects an action from a convex decision set and then an adversary reveals a convex cost and a convex constraint function. The goal of the learner is to select a sequence of actions to minimize both regret and the cumulative constraint violation (CCV) over a horizon of length T. The best-known policy for this problem achieves O( T)regret and O( T)CCV. In this paper, we improve this by trading off regret to achieve substantially smaller CCV. This trade-off is especially important in safety-critical applications, where satisfying the safety constraints is non-negotiable. Specifically, for any bounded convex cost and constraint functions, we propose an online policy that achieves O( dT+Tβ)regret and O(dT1 β)CCV, where dis the dimension of the decision set and β [0,1]is a tunable parameter. We begin with a special case, called the CONSTRAINEDEXPERT problem, where the decision set is a probability simplex and the cost and constraint functions are linear. Leveraging a new adaptive small-loss regret bound, we propose a computationally efficient policy for the CONSTRAINEDEXPERT problem, that attains O( T lnN+Tβ)regret and O(T1 β lnN)CCV for N number of experts.