An adaptive first-order algorithm achieves O(sqrt(T)) alpha-regret and O(B_T log T) + O-tilde(sqrt(T)) resource consumption for online learning with long-term adversarial constraints on alpha-approximately convex functions.
The setting we consider here is the same as the Bandits with Knapsacks (BwK) problem, considered by Immorlica et al
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Online Learning for Approximately-Convex Functions with Long-term Adversarial Constraints
An adaptive first-order algorithm achieves O(sqrt(T)) alpha-regret and O(B_T log T) + O-tilde(sqrt(T)) resource consumption for online learning with long-term adversarial constraints on alpha-approximately convex functions.