Beyond Õ(√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 Õ( √ 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 Õ( √ dT +T β ) regret and Õ(dT 1-β ) CCV, where d is the dimension of the decision set and β ∈ [0, 1] is a tunable parameter. We begin with a special case, called the CONSTRAINED EXPERT 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 CONSTRAINED EXPERT problem, that attains O( √ T ln N + T β ) regret and Õ(T 1-β ln N ) CCV for N number of experts. The original problem is then reduced to the CONSTRAINED EXPERT problem via a covering argument. Finally, with an additional M -smoothness assumption, we propose a computationally efficient first-order policy attaining O( √ M T + T β ) regret and Õ(M T 1-β ) CCV.