  We address the online linear optimization problem with bandit feedback. Our
contribution is twofold. First, we provide an algorithm (based on exponential
weights) with a regret of order $\sqrt{d n \log N}$ for any finite action set
with $N$ actions, under the assumption that the instantaneous loss is bounded
by 1. This shaves off an extraneous $\sqrt{d}$ factor compared to previous
works, and gives a regret bound of order $d \sqrt{n \log n}$ for any compact
set of actions. Without further assumptions on the action set, this last bound
is minimax optimal up to a logarithmic factor. Interestingly, our result also
shows that the minimax regret for bandit linear optimization with expert advice
in $d$ dimension is the same as for the basic $d$-armed bandit with expert
advice. Our second contribution is to show how to use the Mirror Descent
algorithm to obtain computationally efficient strategies with minimax optimal
regret bounds in specific examples. More precisely we study two canonical
action sets: the hypercube and the Euclidean ball. In the former case, we
obtain the first computationally efficient algorithm with a $d \sqrt{n}$
regret, thus improving by a factor $\sqrt{d \log n}$ over the best known result
for a computationally efficient algorithm. In the latter case, our approach
gives the first algorithm with a $\sqrt{d n \log n}$ regret, again shaving off
an extraneous $\sqrt{d}$ compared to previous works.
