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# Online Convex Optimization with Perturbed Constraints: Optimal Rates against Stronger Benchmarks

*Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics*, PMLR 108:2885-2895, 2020.

#### 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.