  We address online linear optimization problems when the possible actions of
the decision maker are represented by binary vectors. The regret of the
decision maker is the difference between her realized loss and the best loss
she would have achieved by picking, in hindsight, the best possible action. Our
goal is to understand the magnitude of the best possible (minimax) regret. We
study the problem under three different assumptions for the feedback the
decision maker receives: full information, and the partial information models
of the so-called "semi-bandit" and "bandit" problems. Combining the Mirror
Descent algorithm and the INF (Implicitely Normalized Forecaster) strategy, we
are able to prove optimal bounds for the semi-bandit case. We also recover the
optimal bounds for the full information setting. In the bandit case we discuss
existing results in light of a new lower bound, and suggest a conjecture on the
optimal regret in that case. Finally we also prove that the standard
exponentially weighted average forecaster is provably suboptimal in the setting
of online combinatorial optimization.
