Breaking the $O(\sqrt{T})$ Cumulative Constraint Violation Barrier while Achieving $O(\sqrt{T})$ 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 $x_t \in \mathcal{X} \subset \mathbb{R}^d$, 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 $Ω(\sqrt{T})$ 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 $d=2$.

Figures (2)

• Figure 1: Regions $S_t$, $S_{t-1}$, half-plane $H_t$, the line $\ell_t$, and intersection points $a_t$, $b_t$.
• Figure 2: Schematics for the proofs of Lemma \ref{['lem:area_decrease']} and \ref{['lem:perim_decrease']}.

Theorems & Definitions (18)

• Definition 4
• Lemma 5
• Theorem 6
• Proposition 7
• Proposition 8
• Remark 1
• Remark 2
• Definition 9
• Theorem 10
• Proposition 11