  Partial monitoring is a general model for sequential learning with limited
feedback formalized as a game between two players. In this game, the learner
chooses an action and at the same time the opponent chooses an outcome, then
the learner suffers a loss and receives a feedback signal. The goal of the
learner is to minimize the total loss. In this paper, we study partial
monitoring with finite actions and stochastic outcomes. We derive a logarithmic
distribution-dependent regret lower bound that defines the hardness of the
problem. Inspired by the DMED algorithm (Honda and Takemura, 2010) for the
multi-armed bandit problem, we propose PM-DMED, an algorithm that minimizes the
distribution-dependent regret. PM-DMED significantly outperforms
state-of-the-art algorithms in numerical experiments. To show the optimality of
PM-DMED with respect to the regret bound, we slightly modify the algorithm by
introducing a hinge function (PM-DMED-Hinge). Then, we derive an asymptotically
optimal regret upper bound of PM-DMED-Hinge that matches the lower bound.
