Breaking the O(√T) Cumulative Constraint Violation Barrier while Achieving O(√T) Static Regret in Constrained Online Convex Optimization

The problem of constrained online convex optimization is considered, where at each round, once a learner commits to an action xt ∈ X subset ℝd, a convex loss function ft and a convex constraint function gt that drives the constraint gt(x)≤ 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 ft and gt for all t ahead of time, and chooses a static optimal action that is feasible with respect to all gt(x)≤ 0. In recent prior work Sinha and Vaze [2024], algorithms with simultaneous regret of O(√T) and CCV of O(√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(√T) for all algorithms that ensure that regret is O(√T) with the worst case input for any d≥ 2. In this paper, we refute this and show that the algorithm of Vaze and Sinha [2025] simultaneously achieves regret of O(√T) regret and CCV of O(T1/3) when d=2.