A Low Complexity Algorithm with $O(\sqrt{T})$ Regret and Constraint Violations for Online Convex Optimization with Long Term Constraints

This paper considers online convex optimization over a complicated constraint set, which typically consists of multiple functional constraints and a set constraint. The conventional Zinkevich's projection based online algorithm (Zinkevich 2013) can be difficult to implement due to the potentially high computation complexity of the projection operation. In this paper, we relax the functional constraints by allowing them to be violated at each round but still requiring them to be satisfied in the long term. This type of relaxed online convex optimization (with long term constraints) was first considered in Mehrdad et. al. (2012). That prior work proposes an algorithm to achieve $O(\sqrt{T})$ regret and $O(T^{3/4})$ constraint violations for general problems and another algorithm to achieve an $O(T^{2/3})$ bound for both regret and constraint violations when the constraint set can be described by a finite number of linear constraints. The current paper proposes a new algorithm that is simple and yields improved performance in comparison to the prior work. The new algorithm achieves an $O(\sqrt{T})$ bound for both regret and constraint violations.