Adaptive Algorithms for Online Convex Optimization with Long-term Constraints

We present an adaptive online gradient descent algorithm to solve online convex optimization problems with long-term constraints, which are constraints that need to be satisfied when accumulated over a finite number of rounds T, but can be violated in intermediate rounds. For some user-defined trade-off parameter βin (0, 1), the proposed algorithm achieves cumulative regret bounds of O(T^maxβ,1_β) and O(T^1_β/2), respectively for the loss and the constraint violations. Our results hold for convex losses, can handle arbitrary convex constraints and rely on a single computationally efficient algorithm. Our contributions improve over the best known cumulative regret bounds of Mahdavi et al. (2012), which are respectively O(T^1/2) and O(T^3/4) for general convex domains, and respectively O(T^2/3) and O(T^2/3) when the domain is further restricted to be a polyhedral set. We supplement the analysis with experiments validating the performance of our algorithm in practice.