Online Convex Optimization with Perturbed Constraints: Optimal Rates against Stronger Benchmarks
Victor Valls, George Iosifidis, Douglas Leith, Leandros Tassiulas
Abstract
This paper studies 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, for example, to model a time-varying budget or time-varying requests in resource allocation problems. Our goal is to design a policy that obtains sublinear regret and satisfies the constraints in the long-term. To this end, we present an online 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. The proposed algorithm obtains optimal rates when $\epsilon = 1/2$, and can compare against a stronger comparator (the set of fixed decisions in hindsight) than previous work.
BibTeX
@InProceedings{pmlr-v108-valls20a,
title = {Online Convex Optimization with Perturbed Constraints: Optimal Rates against Stronger Benchmarks},
author = {Valls, Victor and Iosifidis, George and Leith, Douglas and Tassiulas, Leandros},
booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics},
pages = {2885--2895},
year = {2020},
editor = {Chiappa, Silvia and Calandra, Roberto},
volume = {108},
series = {Proceedings of Machine Learning Research},
month = {26--28 Aug},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v108/valls20a/valls20a.pdf},
url = {https://proceedings.mlr.press/v108/valls20a.html},
abstract = {This paper studies 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, for example, to model a time-varying budget or time-varying requests in resource allocation problems. Our goal is to design a policy that obtains sublinear regret and satisfies the constraints in the long-term. To this end, we present an online 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. The proposed algorithm obtains optimal rates when $\epsilon = 1/2$, and can compare against a stronger comparator (the set of fixed decisions in hindsight) than previous work. }
}