  We consider revenue maximization in online auction/pricing problems. A seller
sells an identical item in each period to a new buyer, or a new set of buyers.
For the online posted pricing problem, we show regret bounds that scale with
the best fixed price, rather than the range of the values. We also show regret
bounds that are almost scale free, and match the offline sample complexity,
when comparing to a benchmark that requires a lower bound on the market share.
These results are obtained by generalizing the classical learning from experts
and multi-armed bandit problems to their multi-scale versions. In this version,
the reward of each action is in a different range, and the regret w.r.t. a
given action scales with its own range, rather than the maximum range.

Computer Science and Game Theory;Data Structures and Algorithms;Machine Learning;Machine Learning