  We study the attainable regret for online linear optimization problems with
bandit feedback, where unlike the full-information setting, the player can only
observe its own loss rather than the full loss vector. We show that the price
of bandit information in this setting can be as large as $d$, disproving the
well-known conjecture that the regret for bandit linear optimization is at most
$\sqrt{d}$ times the full-information regret. Surprisingly, this is shown using
"trivial" modifications of standard domains, which have no effect in the
full-information setting. This and other results we present highlight some
interesting differences between full-information and bandit learning, which
were not considered in previous literature.
