Minimax Regret of Finite Partial-Monitoring Games in Stochastic Environments

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. Assuming that the outcomes are generated in an i.i.d. fashion from an arbitrary and unknown probability distribution, 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, $\widetilde{\Theta}(\sqrt{T}), \Theta(T^{2/3})$, or $\Theta(T)$. We provide a computationally efficient learning algorithm that achieves the minimax regret within logarithmic factor for any game.