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

arXiv: https://arxiv.org/abs/2607.10808 · HTML: https://mlsanity.com/paper/2607.10808/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 15 references that are in the map.
- First posted on arXiv 12 Jul 2026.

## Builds on (most cited first, 9 of 9)

- [Online Convex Optimization with Stochastic Constraints](https://mlsanity.com/paper/1708.03741/) (2017)
- [Adaptive Algorithms for Online Convex Optimization with Long-term Constraints](https://mlsanity.com/paper/1512.07422/) (2015)
- [Online Convex Optimization with Time-Varying Constraints](https://mlsanity.com/paper/1702.04783/) (2017)
- [Online convex optimization for cumulative constraints](https://mlsanity.com/paper/1802.06472/) (2018)
- [Regret and Cumulative Constraint Violation Analysis for Online Convex Optimization with Long Term Constraints](https://mlsanity.com/paper/2106.05135/) (2021)
- [Optimal Algorithms for Online Convex Optimization with Adversarial Constraints](https://mlsanity.com/paper/2310.18955/) (2023)
- [Distributed Online Convex Optimization with Adversarial Constraints: Reduced Cumulative Constraint Violation Bounds under Slater's Condition](https://mlsanity.com/paper/2306.00149/) (2023)
- [A Geometric Approach to Constrained Online Learning](https://mlsanity.com/paper/2605.21107/) (2026)
- [Breaking the O(√T) Cumulative Constraint Violation Barrier while Achieving O(√T) Static Regret in Constrained Online Convex Optimization](https://mlsanity.com/paper/2603.20671/) (2026)

## Abstract (arXiv)

The problem of constrained online convex optimization is considered, where at each round, once a learner commits to an action xt ∈ X subset ℝd, a convex loss function ft and a convex constraint function gt that drives the constraint gt(x)≤ 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 ft and gt for all t ahead of time, and chooses a static optimal action that is feasible with respect to all gt(x)≤ 0. Currently, the best known algorithm is OGD+Projection algorithm of [Vaze and Sinha, 2025] that has simultaneous regret of O(√T) and CCV of O(T1/3) for d=2 [Balasundaram et al., 2026], and simultaneous regret of O(√T) and CCV of O(√T) for any d [Sarkar and Sinha, 2026]. In this paper, we show that the CCV of the OGD+Projection algorithm is Omega (T^fracd-12d). This is the first such lower bound result.