Beyond $\tilde{O}(\sqrt{T})$ 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 $O(\sqrt{T})$ regret and $\tilde{O}(\sqrt{T})$ 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 $\tilde{O}(\sqrt{dT}+T^{\beta })$ regret and $\tilde{O}(d{T}^{1-\beta })$ CCV, where $d$ is the dimension of the decision set and $\beta \in 0,1$ 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 $O(\sqrt{T\ln N}+T^{\beta })$ regret and $\tilde{O}(T^{1-\beta }\ln N)$ 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 $O(\sqrt{MT}+T^{\beta })$ regret and $\tilde{O}(M{T}^{1-\beta })$ CCV.