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

2026-03-21

AI Summary

This work addresses constrained online convex optimization, where both the loss and constraint functions are revealed sequentially and future information is unknown. The goal is to simultaneously minimize static regret and cumulative constraint violation (CCV). For problems in dimension $d \geq 2$, the study breaks the long-standing $\Omega(\sqrt{T})$ lower bound on CCV, previously believed to be tight. Building upon the algorithmic framework of Vaze and Sinha [2025] and integrating tools from convex analysis with adaptive constraint-handling techniques, the authors establish the first joint guarantee achieving $O(\sqrt{T})$ static regret and $O(T^{1/3})$ CCV when $d = 2$, markedly improving upon existing theoretical bounds.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $d=2$.

Problem

• constrained online convex optimization
• static regret
• cumulative constraint violation
• regret minimization
• constraint violation

Innovation

• constrained online convex optimization
• static regret
• cumulative constraint violation
• sublinear violation
• O(T^{1/3})