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Partial monitoring is a general model for sequential learning with limitedfeedback formalized as a game between two players. In this game, the learnerchooses an action and at the same time the opponent chooses an outcome, thenthe learner suffers a loss and receives a feedback signal. The goal of thelearner is to minimize the total loss. In this paper, we study partialmonitoring with finite actions and stochastic outcomes. We derive a logarithmicdistribution-dependent regret lower bound that defines the hardness of theproblem. Inspired by the DMED algorithm (Honda and Takemura, 2010) for themulti-armed bandit problem, we propose PM-DMED, an algorithm that minimizes thedistribution-dependent regret. PM-DMED significantly outperformsstate-of-the-art algorithms in numerical experiments. To show the optimality ofPM-DMED with respect to the regret bound, we slightly modify the algorithm byintroducing a hinge function (PM-DMED-Hinge). Then, we derive an asymptoticallyoptimal regret upper bound of PM-DMED-Hinge that matches the lower bound.

Regret Lower Bound and Optimal Algorithm in Finite Stochastic Partial Monitoring