Breaking the $O(sqrt{T})$ Cumulative Constraint Violation Barrier while Achieving $O(sqrt{T})$ Static Regret in Constrained Online Convex Optimization

arXiv:2603.20671v1 Announce Type: cross
Abstract: The problem of constrained online convex optimization is considered, where at each round, once a learner commits to an action $x_t in mathcal{X} subset mathbb{R}^d$, a convex loss function $f_t$ and a convex constraint function $g_t$ that drives the constraint $g_t(x)le 0$ are revealed. The objective is to simultaneously minimize the static regret and cumulative constraint violation (CCV) compared to the benchmark that knows the loss functions and constraint functions $f_t$ and $g_t$ for all $t$ ahead of time, and chooses a static optimal action that is feasible with respect to all $g_t(x)le 0$. In recent prior work Sinha and Vaze [2024], algorithms with simultaneous regret of $O(sqrt{T})$ and CCV of $O(sqrt{T})$ or (CCV of $O(1)$ in specific cases Vaze and Sinha [2025], e.g. when $d=1$) have been proposed. It is widely believed that CCV is $Omega(sqrt{T})$ for all algorithms that ensure that regret is $O(sqrt{T})$ with the worst case input for any $dge 2$. In this paper, we refute this and show that the algorithm of Vaze and Sinha [2025] simultaneously achieves regret of $O(sqrt{T})$ regret and CCV of $O(T^{1/3})$ when $d=2$.