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

TL;DR

This paper introduces an online algorithm for Network Revenue Management with continuous-valued arrivals, achieving logarithmic regret without non-degeneracy assumptions, advancing theoretical understanding of such models.

Contribution

It presents the first logarithmic regret bounds for NRM with continuous distributions under minimal assumptions, using novel bounding and relaxation techniques.

Findings

Achieves $O( ext{log}^2 T)$ regret with bounded density assumption.

Improves to $O( ext{log} T)$ regret with second-order growth assumption.

Introduces new methods for bounding regret and dual convergence.

Abstract

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".