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

Jiashuo Jiang, Will Ma, Jiawei Zhang

TL;DR

This work resolves long-standing questions about regret in online network revenue management when demands have indiscrete distributions by introducing a semi-fluid relaxation that blends offline optimality with fluid approximations. It proves $O(\log^2 T)$ regret under densities bounded away from zero and strengthens to $O(\log T)$ with a second-order growth condition and bounded densities, all without non-degeneracy assumptions. The authors further connect the framework to price-based NRM via virtual valuations, and provide extensive theoretical machinery around myopic regret, dual convergence, and perturbation analysis. Empirical results corroborate the theoretical gains, showing substantial improvements over classic bid-price and dual-update policies, especially as horizon length and problem size grow, highlighting the practical impact for large-scale revenue management systems.

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".

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

TL;DR

This work resolves long-standing questions about regret in online network revenue management when demands have indiscrete distributions by introducing a semi-fluid relaxation that blends offline optimality with fluid approximations. It proves regret under densities bounded away from zero and strengthens to with a second-order growth condition and bounded densities, all without non-degeneracy assumptions. The authors further connect the framework to price-based NRM via virtual valuations, and provide extensive theoretical machinery around myopic regret, dual convergence, and perturbation analysis. Empirical results corroborate the theoretical gains, showing substantial improvements over classic bid-price and dual-update policies, especially as horizon length and problem size grow, highlighting the practical impact for large-scale revenue management systems.

Abstract

We study the classical Network Revenue Management (NRM) problem with accept/reject decisions and 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 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 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".
Paper Structure (20 sections, 24 theorems, 313 equations, 1 figure, 3 algorithms)

This paper contains 20 sections, 24 theorems, 313 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Logarithmic Regret for NRM with Discrete Demands and Continuous Rewards
  3. Further Related Work
  4. Problem Formulation and Our Approach
  5. $O(\log^2 T)$ Regret with Lower-bounded Densities
  6. $O(\log T)$ Regret with Bounded Densities and Second-order Growth Assumption
  7. Policy with Log-squared Regret
  8. General Description of Our Approach
  9. Semi-fluid Relaxation
  10. Policy and Regret Analysis
  11. Policy with Logarithmic Regret under Second-order Growth
  12. Decomposition of the Myopic Regret
  13. Reduction to Dual Convergence
  14. Bound on Dual Convergence and Policy
  15. Numerical Performances of Our Algorithms
  16. ...and 5 more sections

Key Result

Lemma 1

For any feasible online policy $\pi$, the regret is upper bounded by where the myopic term $\text{Myopic}_t(\pi,\tilde{\bm{c}}_t^{\pi})$ is defined in def:myopicregret.

Figures (1)

  • Figure 1: The comparison between the four policies, fixed bid price policy, dual update policy, \ref{['alg:M1']}, and \ref{['alg:MLog']}.

Theorems & Definitions (35)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Example 1
  • Example 2
  • ...and 25 more