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

Abstract

The problem of constrained online convex optimization is considered, where at each round, once a learner commits to an action x_t X R^d, a convex loss function f_t and a convex constraint function g_t that drives the constraint g_t(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 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) 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 Ω(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(T^1/3) when d=2.

Reproductions