# On Dynamic Regret and Constraint Violations in Constrained Online Convex Optimization

arXiv: https://arxiv.org/abs/2301.09808 · HTML: https://mlsanity.com/paper/2301.09808/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 25 references that are in the map.
- First posted on arXiv 24 Jan 2023.

## Builds on (most cited first, 11 of 11)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Non-stationary Stochastic Optimization](https://mlsanity.com/paper/1307.5449/) (2013)
- [Adaptive Online Learning in Dynamic Environments](https://mlsanity.com/paper/1810.10815/) (2018)
- [Tracking Slowly Moving Clairvoyant: Optimal Dynamic Regret of Online Learning with True and Noisy Gradient](https://mlsanity.com/paper/1605.04638/) (2016)
- [Improved Dynamic Regret for Non-degenerate Functions](https://mlsanity.com/paper/1608.03933/) (2016)
- [Dynamical Models and Tracking Regret in Online Convex Programming](https://mlsanity.com/paper/1301.1254/) (2013)
- [Bandit Convex Optimization in Non-stationary Environments](https://mlsanity.com/paper/1907.12340/) (2019)
- [Improved Analysis for Dynamic Regret of Strongly Convex and Smooth Functions](https://mlsanity.com/paper/2006.05876/) (2020)
- [Minimizing Dynamic Regret and Adaptive Regret Simultaneously](https://mlsanity.com/paper/2002.02085/) (2020)
- [Bandit Convex Optimization for Scalable and Dynamic IoT Management](https://mlsanity.com/paper/1707.09060/) (2017)
- [Simultaneously Achieving Sublinear Regret and Constraint Violations for Online Convex Optimization with Time-varying Constraints](https://mlsanity.com/paper/2111.07707/) (2021)

## Built on by (most cited first, 4 of 4)

- [Optimal Algorithms for Online Convex Optimization with Adversarial Constraints](https://mlsanity.com/paper/2310.18955/) (2023)
- [Online Convex Optimization with Switching Cost and Delayed Gradients](https://mlsanity.com/paper/2310.11880/) (2023)
- [Dynamic Regret Bounds for Online Omniprediction with Long Term Constraints](https://mlsanity.com/paper/2510.07266/) (2025)
- [Multi-Objective textitmin-max Online Convex Optimization](https://mlsanity.com/paper/2510.13560/) (2025)

## Abstract (arXiv)

A constrained version of the online convex optimization (OCO) problem is considered. With slotted time, for each slot, first an action is chosen. Subsequently the loss function and the constraint violation penalty evaluated at the chosen action point is revealed. For each slot, both the loss function as well as the function defining the constraint set is assumed to be smooth and strongly convex. In addition, once an action is chosen, local information about a feasible set within a small neighborhood of the current action is also revealed. An algorithm is allowed to compute at most one gradient at its point of choice given the described feedback to choose the next action. The goal of an algorithm is to simultaneously minimize the dynamic regret (loss incurred compared to the oracle's loss) and the constraint violation penalty (penalty accrued compared to the oracle's penalty). We propose an algorithm that follows projected gradient descent over a suitably chosen set around the current action. We show that both the dynamic regret and the constraint violation is order-wise bounded by the it path-length, the sum of the distances between the consecutive optimal actions. Moreover, we show that the derived bounds are the best possible.