Partial Monitoring - Classification, Regret Bounds, and Algorithms

In a partial monitoring game, the learner repeatedly chooses an action, the environment responds with an outcome, and then the learner suffers a loss and receives a feedback signal, both of which are fixed functions of the action and the outcome. The goal of the learner is to minimize his regret, which is the difference between his total cumulative loss and the total loss of the best fixed action in hindsight. In this paper we characterize the minimax regret of any partial monitoring game with finitely many actions and outcomes. It turns out that the minimax regret of any such game is either zero or scales as T1/2, T2/3, or T up to constants and logarithmic factors. We provide computationally efficient learning algorithms that achieve the minimax regret within a logarithmic factor for any game. In addition to the bounds on the minimax regret, if we assume that the outcomes are generated in an i.i.d. fashion, we prove individual upper bounds on the expected regret.