  We resolve the long-standing "impossible tuning" issue for the classic expert
problem and show that, it is in fact possible to achieve regret
$O\left(\sqrt{(\ln d)\sum_t \ell_{t,i}^2}\right)$ simultaneously for all expert
$i$ in a $T$-round $d$-expert problem where $\ell_{t,i}$ is the loss for expert
$i$ in round $t$. Our algorithm is based on the Mirror Descent framework with a
correction term and a weighted entropy regularizer. While natural, the
algorithm has not been studied before and requires a careful analysis. We also
generalize the bound to $O\left(\sqrt{(\ln d)\sum_t
(\ell_{t,i}-m_{t,i})^2}\right)$ for any prediction vector $m_t$ that the
learner receives, and recover or improve many existing results by choosing
different $m_t$. Furthermore, we use the same framework to create a master
algorithm that combines a set of base algorithms and learns the best one with
little overhead. The new guarantee of our master allows us to derive many new
results for both the expert problem and more generally Online Linear
Optimization.
