Online Learning with Sample Path Constraints

Shie Mannor, John N. Tsitsiklis, Jia Yuan Yu; 10(Mar):569--590, 2009.

Abstract

We study online learning where a decision maker interacts with Nature with the objective of maximizing her long-term average reward subject to some sample path average constraints. 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. Using a calibrated forecasting rule, we provide an explicit strategy that attains this convex hull. We also measure the performance of heuristic methods based on non-calibrated forecasters in experiments involving a CPU power management problem.

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