Improved Guarantees for Constrained Online Convex Optimization via Self-Contraction

2026-05-20

AI Summary

This work addresses online convex optimization under adversarial constraints, aiming to simultaneously minimize static regret and cumulative constraint violation. The authors propose a novel projection-based online algorithm that leverages the geometric properties of self-contracted curves, making decisions before observing the loss and constraint functions at each round. In the strongly convex setting, the algorithm achieves an $O(\log T)$ bound on both static regret and cumulative constraint violation—the latter improving upon the previous best-known rate of $O(\sqrt{T \log T})$. For general convex objectives, it attains $O(\sqrt{T})$ regret and $O(\sqrt{T})$ cumulative constraint violation, matching the optimal order in both measures.

Abstract

We consider Constrained Online Convex Optimization (COCO) with adversarially chosen constraints. At each round, the learner chooses an action before observing the loss and constraint function for that round. The goal is to achieve small static regret against the best point satisfying all constraints while also controlling cumulative constraint violation ($\mathsf{CCV}$). For strongly convex losses, state-of-the-art algorithms achieve $O(\log T)$ regret and $O(\sqrt{T \log T})$ $\mathsf{CCV}.$ The corresponding best-known bounds for convex losses is $O(\sqrt{T})$ regret and $O(\sqrt{T} \log T)$ $\mathsf{CCV}$. In this paper, we give a simple projection-based algorithm that simultaneously achieves $O(\log T)$ regret and $O(\log T)$ $\mathsf{CCV}$ for strongly-convex losses, yielding an exponential improvement in the $\mathsf{CCV}$. For the convex losses, our algorithm improves the $\mathsf{CCV}$ to $O(\sqrt{T})$ while maintaining the optimal $O(\sqrt{T})$ regret. The key to our improvement is a recent geometric result for self-contracted curves, which may be of independent interest.

Problem

• Constrained Online Convex Optimization
• cumulative constraint violation
• static regret
• strongly convex losses
• adversarial constraints

Innovation

• Constrained Online Convex Optimization
• Self-Contracted Curves
• Cumulative Constraint Violation
• Strongly Convex Losses
• Projection-Based Algorithm