  We consider online learning problems where the aim is to achieve regret which
is efficient in the sense that it is the same order as the lowest regret
amongst K experts. This is a substantially stronger requirement that achieving
$O(\sqrt{n})$ or $O(\log n)$ regret with respect to the best expert and
standard algorithms are insufficient, even in easy cases where the regrets of
the available actions are very different from one another. We show that a
particular lazy form of the online subgradient algorithm can be used to achieve
minimal regret in a number of "easy" regimes while retaining an $O(\sqrt{n})$
worst-case regret guarantee. We also show that for certain classes of problem
minimal regret strategies exist for some of the remaining "hard" regimes.
