# Online Convex Optimization with Perturbed Constraints

## TL;DR

This paper introduces a primal-dual proximal gradient algorithm for online convex optimization with time-varying, unknown perturbations in constraints, achieving sublinear regret and constraint violation with adaptive learning rates.

## Contribution

It proposes a novel algorithm that balances regret and constraint violation, handling unknown, non-i.i.d. perturbations with adaptive learning rates.

## Key findings

- Achieves $O(T^ hreshold)$ regret and $O(T^ hreshold)$ constraint violation.
- Defines regret with a time-varying set of best fixed decisions.
- Supports any time horizon with adaptive learning rates.

## Abstract

This paper addresses Online Convex Optimization (OCO) problems where the constraints have additive perturbations that (i) vary over time and (ii) are not known at the time to make a decision. Perturbations may not be i.i.d. generated and can be used to model a time-varying budget or commodity in resource allocation problems. The problem is to design a policy that obtains sublinear regret while ensuring that the constraints are satisfied on average. To solve this problem, we present a primal-dual proximal gradient algorithm that has $O(T^\epsilon \vee T^{1-\epsilon})$ regret and $O(T^\epsilon)$ constraint violation, where $\epsilon \in [0,1)$ is a parameter in the learning rate. Our results match the bounds of previous work on OCO with time-varying constraints when $\epsilon = 1/2$; however, we (i) define the regret using a time-varying set of best fixed decisions; (ii) can balance between regret and constraint violation; and (iii) use an adaptive learning rate that allows us to run the algorithm for any time horizon.