$\tilde{O}(\sqrt{T})$ Regret and Polylogarithmic Constraint Violation for COCO

We study constrained online convex optimization with adversarial convex costs and constraints ($\mathsf{COCO}$). At each round $t\in[T]$, a learner selects $x_t$ from a $d$-dimensional convex decision set $X$, after which an adaptive adversary reveals a convex cost function $f_t$ and constraint function $g_t$. Consequently, the learner incurs a cost of $f_t(x_t)$ and a constraint violation of $\max\{0,g_t(x_t)\}$, and aims to simultaneously minimize regret and cumulative constraint violation ($\mathsf{CCV}$) over the entire horizon. Existing algorithms achieve $O(\sqrt{T})$ regret and $\widetilde O(\sqrt{T})$ $\mathsf{CCV}$. We show that an online policy can achieve $O(\sqrt{T\log T})$ regret and $O(\log^2 T)$ $\mathsf{CCV}$, reducing the $\mathsf{CCV}$ from polynomial to polylogarithmic while retaining near-optimal regret. When all cost functions are either $α$-exp-concave or $β$-strongly convex, the algorithm achieves $O(\log T)$ regret with the same $\mathsf{CCV}$ bound as above. 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, Grünbaum'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 $O(\log^2 T)$ on the $\mathsf{CCV}$.