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This paper considers online convex optimization over a complicated constraintset, which typically consists of multiple functional constraints and a setconstraint. The conventional online projection algorithm (Zinkevich, 2003) canbe difficult to implement due to the potentially high computation complexity ofthe projection operation. In this paper, we relax the functional constraints byallowing them to be violated at each round but still requiring them to besatisfied in the long term. This type of relaxed online convex optimization(with long term constraints) was first considered in Mahdavi 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 algorithmto achieve an $O(T^{2/3})$ bound for both regret and constraint violations whenthe constraint set can be described by a finite number of linear constraints. Arecent extension in \citet{Jenatton16ICML} can achieve$O(T^{\max\{\theta,1-\theta\}})$ regret and $O(T^{1-\theta/2})$ constraintviolations where $\theta\in (0,1)$. The current paper proposes a new simplealgorithm that yields improved performance in comparison to prior works. Thenew algorithm achieves an $O(\sqrt{T})$ regret bound with $O(1)$ constraintviolations.

A Low Complexity Algorithm with $O(\sqrt{T})$ Regret and $O(1)$ Constraint Violations for Online Convex Optimization with Long Term Constraints