Dynamic resource allocation: The geometry and robustness of constant regret

We study a family of dynamic resource allocation problems, wherein requests of different types arrive over time and are accepted or rejected. Each request type is characterized by its reward and the resources it consumes. We consider an algorithm, called BudgetRatio, that solves an intuitive packing linear program and accepts requests “scored” as sufficiently valuable by the LP. Our analysis method is geometric and focuses on the evolution of the remaining inventory—hence of the LP that drives BudgetRatio—as a stochastic process. The analysis requires a detailed characterization of the parametric structure of the packing LP, which is of independent interest. We prove that (i) BudgetRatio achieves constant regret relative to the offline (full information) upper bound in the presence of (slow) inventory restock and request queues. (ii) By tuning the algorithm’s parameters, we simultaneously achieve near-maximal reward and near-minimal holding cost. (iii) Within explicitly identifiable bounds, the algorithms regret is robust to mis-specification of the model parameters. This gives bounds for the “bandits” version where the parameters have to be learned. (iv) The algorithm has a natural interpretation as a generalized bid-price algorithm.