$O(\sqrt{T})$ Static Regret and Instance Dependent Constraint Violation for Constrained Online Convex Optimization

TL;DR

This paper introduces an online convex optimization algorithm that achieves an $O(\sqrt{T})$ static regret and an instance-dependent constraint violation bound, improving adaptability to specific problem structures.

Contribution

The paper presents a new algorithm for constrained online convex optimization with static regret of $O(\sqrt{T})$ and an instance-dependent constraint violation bound, leveraging geometric properties of constraints.

Findings

Achieves $O(\sqrt{T})$ static regret.

Provides an instance-dependent bound on cumulative constraint violation.

Outperforms previous universal bounds by exploiting geometric properties.

Abstract

The constrained version of the standard online convex optimization (OCO) framework, called COCO is considered, where on every round, a convex cost function and a convex constraint function are revealed to the learner after it chooses the action for that round. The objective is to simultaneously minimize the static regret and cumulative constraint violation (CCV). An algorithm is proposed that guarantees a static regret of $O(\sqrt{T})$ and a CCV of $\min\{\cV, O(\sqrt{T}\log T) \}$, where $\cV$ depends on the distance between the consecutively revealed constraint sets, the shape of constraint sets, dimension of action space and the diameter of the action space. For special cases of constraint sets, $\cV=O(1)$. Compared to the state of the art results, static regret of $O(\sqrt{T})$ and CCV of $O(\sqrt{T}\log T)$, that were universal, the new result on CCV is instance dependent, which is derived by exploiting the geometric properties of the constraint sets.