Bandits with Knapsacks beyond the Worst Case

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 Agrawal and Devanur (2014), providing new analyses thereof. * The initial version, titled "Advances in Bandits with Knapsacks", was published on arxiv.org in Jan'20. The present version (since Dec'20) improves both upper and lower bounds, deriving Theorem 3.2(ii) and Theorem 4.2. Moreover, it simplifies the algorithm and analysis in the main result, and fixes several issues in the lower bounds. The latest version (Dec'21