tildeO(√T) Regret and Polylogarithmic Constraint Violation for COCO

We study constrained online convex optimization with adversarial convex losses and constraints (mathsfCOCO). At each round (t∈[T]), a learner selects (xt) from a (d)-dimensional convex decision set (mathcal X), after which an adaptive adversary reveals a convex cost function (ft) and constraint function (gt). Consequently, the learner incurs cost (ft(xt)) and constraint violation (max0,gt(xt)), and aims to simultaneously minimize regret and cumulative constraint violation (mathsfCCV) over the entire horizon. Existing algorithms achieve (O(√T)) regret and (widetilde O(√T)) mathsfCCV. We show that an online policy can achieve (O(√Tlog T)) regret and (O(log2 T)) mathsfCCV, reducing the mathsfCCV from polynomial to polylogarithmic while retaining near-optimal regret. Our approach combines continuous Hedge with elimination on shrinking feasible sets. The key observation is that whenever the mean of the Hedge distribution violates a constraint, Gr"unbaum's inequality guarantees that a constant fraction of the Hedge probability mass is eliminated. We use an adaptive learning-rate schedule and a potential function coupling the surviving volume with the learning rate to convert this probability-mass reduction into a bound of (O(log2 T)) on the mathsfCCV.