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

TL;DR

This paper challenges the prevailing belief that constrained online convex optimization algorithms cannot achieve sublinear cumulative constraint violation while maintaining optimal regret, by presenting an algorithm with improved bounds in two-dimensional settings.

Contribution

The paper introduces an algorithm that achieves $O(rac{1}{3})$-power cumulative constraint violation alongside $O(rac{1}{2})$-power regret in 2D constrained online convex optimization, refuting prior conjectures.

Findings

Achieves $O(rac{1}{3})$-power CCV in 2D

Maintains $O(rac{1}{2})$-power static regret

Refutes the belief that CCV must be $oldsymbol{ ilde{ ext{O}}}(oldsymbol{ ext{T}})$ for $O( ext{T}^{1/2})$ regret

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 $d\ge 2$. 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 $d=2$.