Online Learning with Constraints

We study online learning where the objective of the decision maker is to maximize her average long-term reward given that some average constraints are satisfied along the sample path. We define the reward-in-hindsight as the highest reward the decision maker could have achieved, while satisfying the constraints, had she known Nature’s choices in advance. We show that in general the reward-in-hindsight is not attainable. The convex hull of the reward-in-hindsight function is, however, attainable. For the important case of a single constraint the convex hull turns out to be the highest attainable function. We further provide an explicit strategy that attains this convex hull using a calibrated forecasting rule.