  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 the only (necessary)
assumption being that the probability densities are bounded away from 0. We
derive a second result that achieves $O(\log T)$ regret under an additional
assumption of second-order growth. To our knowledge, these are the first
results achieving logarithmic-level regret in an NRM model with continuous
values that do not require any kind of ``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''.
