$\tilde{O}(\sqrt{T})$ Regret and Polylogarithmic Constraint Violation for COCO

2026-10-02

AI Summary

This study addresses the problem of excessive cumulative constraint violation in online convex optimization with constraints under adversarial convex losses. To this end, it proposes a coupled framework integrating the continuous Hedge algorithm with a shrinking feasible set elimination strategy. The approach controls regret via adaptive learning rate scheduling and potential function techniques, while leveraging Grünbaum's inequality to precisely analyze probability mass reduction for suppressing constraint violations. This work makes the first contribution to reducing cumulative constraint violation from polynomial to poly-logarithmic order. Theoretically, the proposed algorithm simultaneously achieves a near-optimal regret bound of $O(\sqrt{T \log T})$ and a cumulative constraint violation of $O(\log^2 T)$, representing a significant breakthrough in balancing optimization performance with constraint satisfaction.

Abstract

We study constrained online convex optimization with adversarial convex losses and constraints ($\mathsf{COCO}$). At each round \(t\in[T]\), a learner selects \(x_t\) from a \(d\)-dimensional convex decision set \(\mathcal X\), after which an adaptive adversary reveals a convex cost function \(f_t\) and constraint function \(g_t\). Consequently, the learner incurs cost \(f_t(x_t)\) and constraint violation \(\max\{0,g_t(x_t)\}\), and aims to simultaneously minimize regret and cumulative constraint violation ($\mathsf{CCV}$) over the entire horizon. Existing algorithms achieve \(O(\sqrt{T})\) regret and \(\widetilde O(\sqrt{T})\) $\mathsf{CCV}$. We show that an online policy can achieve \(O(\sqrt{T\log T})\) regret and \(O(\log^2 T)\) $\mathsf{CCV}$, reducing the $\mathsf{CCV}$ from polynomial to polylogarithmic while retaining near-optimal regret. Our approach combines continuous Hedge with elimination on shrinking feasible sets. The key observation is that whenever the mean of the Hedge distribution violates a constraint, Gr\"unbaum's inequality guarantees that a constant fraction of the Hedge probability mass is eliminated. We use an adaptive learning-rate schedule and a potential function coupling the surviving volume with the learning rate to convert this probability-mass reduction into a bound of \(O(\log^2 T)\) on the $\mathsf{CCV}$.

Problem

• Constrained Online Convex Optimization
• Regret Minimization
• Cumulative Constraint Violation
• Adversarial Learning

Innovation

• Constrained Online Convex Optimization
• Cumulative Constraint Violation
• Continuous Hedge
• Grünbaum's Inequality
• Adaptive Learning Rate