  Bandits with Knapsacks (BwK) is a general model for multi-armed bandits under
supply/budget constraints. While worst-case regret bounds for BwK are
well-understood, we present three results that go beyond the worst-case
perspective. First, we provide upper and lower bounds which amount to a full
characterization for logarithmic, instance-dependent regret rates. Second, we
consider "simple regret" in BwK, which tracks algorithm's performance in a
given round, and prove that it is small in all but a few rounds. Third, we
provide a general "reduction" from BwK to bandits which takes advantage of some
known helpful structure, and apply this reduction to combinatorial
semi-bandits, linear contextual bandits, and multinomial-logit bandits. Our
results build on the BwK algorithm from \citet{AgrawalDevanur-ec14}, providing
new analyses thereof.
