Tight Bounds for Online Convex Optimization with Adversarial Constraints
<|reference_start|>Tight Bounds 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 small cumulative constraint violation (CCV) against an adaptive adversary. A long-standing open question in COCO is whether an online policy can simultaneously achieve $O(\sqrt{T})$ regret and $O(\sqrt{T})$ CCV without any restrictive assumptions. For the first time, we answer this in the affirmative and show that an online policy can simultaneously achieve $O(\sqrt{T})$ regret and $\tilde{O}(\sqrt{T})$ CCV. We establish this result 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.<|reference_end|>
@article{sinha2024tight,
    title={Tight Bounds for Online Convex Optimization with Adversarial Constraints},
    author={Abhishek Sinha and Rahul Vaze},
    journal={arXiv preprint arXiv:2405.09296},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.09296},
    primaryClass={cs.LG math.OC}
}