  We consider the Scale-Free Adversarial Multi Armed Bandits(MAB) problem. At
the beginning of the game, the player only knows the number of arms $n$. It
does not know the scale and magnitude of the losses chosen by the adversary or
the number of rounds $T$. In each round, it sees bandit feedback about the loss
vectors $l_1,\dots, l_T \in \mathbb{R}^n$. The goal is to bound its regret as a
function of $n$ and norms of $l_1,\dots, l_T$. We design a bandit Follow The
Regularized Leader (FTRL) algorithm, that uses an adaptive learning rate and
give two different regret bounds, based on the exploration parameter used. With
non-adaptive exploration, our algorithm has a regret of
$\tilde{\mathcal{O}}(\sqrt{nL_2} + L_\infty\sqrt{nT})$ and with adaptive
exploration, it has a regret of $\tilde{\mathcal{O}}(\sqrt{nL_2} +
L_\infty\sqrt{nL_1})$. Here $L_\infty = \sup_t \| l_t\|_\infty$, $L_2 =
\sum_{t=1}^T \|l_t\|_2^2$, $L_1 = \sum_{t=1}^T \|l_t\|_1$ and the
$\tilde{\mathcal{O}}$ notation suppress logarithmic factors. These are the
first MAB bounds that adapt to the $\|\cdot\|_2$, $\|\cdot\|_1$ norms of the
losses. The second bound is the first data-dependent scale-free MAB bound as
$T$ does not directly appear in the regret. We also develop a new technique for
obtaining a rich class of local-norm lower-bounds for Bregman Divergences. This
technique plays a crucial role in our analysis for controlling the regret when
using importance weighted estimators of unbounded losses. This technique could
be of independent interest.
