Toward a classification of finite partial-monitoring games

András Antos, Gábor Bartók, Dávid Pál, Csaba Szepesvári

Abstract: Partial-monitoring games constitute a mathematical framework for sequential decision making problems with imperfect feedback: the learner repeatedly chooses an action, the opponent 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 total cumulative loss. We make progress toward the classification of these games based on their minimax expected regret. Namely, we classify almost all games with two outcomes and a finite number of actions: we show that their minimax expected regret is either zero, Θ˜(T)<math><mover accent="true" is="true"><mrow is="true"><mi is="true">Θ</mi></mrow><mrow is="true"><mo is="true">˜</mo></mrow></mover><mrow is="true"><mo is="true">(</mo><msqrt is="true"><mrow is="true"><mi is="true">T</mi></mrow></msqrt><mo is="true">)</mo></mrow></math>, Θ(T2/3)<math><mi is="true">Θ</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">T</mi></mrow><mrow is="true"><mn is="true">2</mn><mo is="true">/</mo><mn is="true">3</mn></mrow></msup><mo is="true">)</mo></mrow></math>, or Θ(T)<math><mi is="true">Θ</mi><mrow is="true"><mo is="true">(</mo><mi is="true">T</mi><mo is="true">)</mo></mrow></math>, and we give a simple and efficiently computable classification of these four classes of games. Our hope is that the result can serve as a stepping stone toward classifying all finite partial-monitoring games.