Degeneracy is OK: Logarithmic Regret for Network Revenue Management with Indiscrete Distributions

We study the classical Network Revenue Management (NRM) problem with accept/reject decisions and $T$ IID arrivals. We consider a distributional form where each arrival must fall under a finite number of possible categories, each with a deterministic resource consumption vector, but a random value distributed continuously over an interval. We develop an online algorithm that achieves $O(\log^2 T)$ regret under this model, with no further assumptions. We develop another online algorithm that achieves an improved $O(\log T)$ regret, with only a second-order growth assumption. To our knowledge, these are the first results achieving logarithmic-level regret in a continuous-distribution NRM model without further ``non-degeneracy'' assumptions. Our results are achieved via new techniques including: a new method of bounding myopic regret, a ``semi-fluid'' relaxation of the offline allocation, and an improved bound on the ``dual convergence''.