Lower Bound on the Cumulative Constrained Violation for the OGD+Projection algorithm for Constrained Online Convex Optimization (COCO)

2026-07-12

AI Summary

This work investigates the problem of simultaneously minimizing static regret and cumulative constraint violation (CCV) in constrained online convex optimization. Focusing on the classical Online Gradient Descent with Projection (OGD+Projection) algorithm, the study establishes—for the first time—a theoretical lower bound on CCV of $\Omega(T^{(d-1)/(2d)})$ for any dimension $d$. This result closes a significant gap in the existing theoretical analysis and reveals fundamental performance limitations of the algorithm in high-dimensional settings. By precisely characterizing this trade-off, the paper provides crucial theoretical insights into the inherent tension between regret minimization and constraint satisfaction in constrained online learning.

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 $g_t(x)\le 0$. Currently, the best known algorithm is OGD+Projection algorithm of [Vaze and Sinha, 2025] that has simultaneous regret of $O(\sqrt{T})$ and CCV of $O(T^{1/3})$ for $d=2$ [Balasundaram et al., 2026], and simultaneous regret of $O(\sqrt{T})$ and CCV of $O(\sqrt{T})$ for any $d$ [Sarkar and Sinha, 2026]. In this paper, we show that the CCV of the OGD+Projection algorithm is $Ω(T^{\frac{d-1}{2d}})$. This is the first such lower bound result.

Problem

• constrained online convex optimization
• cumulative constraint violation
• lower bound
• OGD+Projection algorithm
• static regret

Innovation

• constrained online convex optimization
• cumulative constraint violation
• lower bound
• OGD+Projection
• online learning