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Near-Optimal Regret-Queue Length Tradeoff in Online Learning for Two-Sided Markets

Zixian Yang, Sushil Mahavir Varma, Lei Ying

TL;DR

The paper studies pricing and matching in a two-sided queueing market with unknown demand and supply curves, proposing a parameter-agnostic online policy that achieves a near-optimal Pareto frontier between profit regret and queue lengths. It combines a MaxWeight-like matching rule with a probabilistic two-price learning scheme, leveraging a two-point zero-order method and a bounded-bisection pricing loop to learn prices while controlling delays. The main theoretical contribution is a $\tilde{O}(T^{1-\gamma})$ regret, $\tilde{O}(T^{\gamma/2})$ AvgQLen, and $O(T^{\gamma})$ MaxQLen bound for $\gamma\in[0,1/6]$, with a near-matching lower bound in a single-link case, establishing near-optimality within a broad policy class. The numerical results corroborate the theoretical trade-offs and show substantial queue-length improvements over prior learning-based approaches, with scalable performance across multi-link settings.

Abstract

We study a two-sided market, wherein, price-sensitive heterogeneous customers and servers arrive and join their respective queues. A compatible customer-server pair can then be matched by the platform, at which point, they leave the system. Our objective is to design pricing and matching algorithms that maximize the platform's profit, while maintaining reasonable queue lengths. As the demand and supply curves governing the price-dependent arrival rates may not be known in practice, we design a novel online-learning-based pricing policy and establish its near-optimality. In particular, we prove a tradeoff among three performance metrics: $\tilde{O}(T^{1-γ})$ regret, $\tilde{O}(T^{γ/2})$ average queue length, and $\tilde{O}(T^γ)$ maximum queue length for $γ\in (0, 1/6]$, significantly improving over existing results [1]. Moreover, barring the permissible range of $γ$, we show that this trade-off between regret and average queue length is optimal up to logarithmic factors under a class of policies, matching the optimal one as in [2] which assumes the demand and supply curves to be known. Our proposed policy has two noteworthy features: a dynamic component that optimizes the tradeoff between low regret and small queue lengths; and a probabilistic component that resolves the tension between obtaining useful samples for fast learning and maintaining small queue lengths.

Near-Optimal Regret-Queue Length Tradeoff in Online Learning for Two-Sided Markets

TL;DR

The paper studies pricing and matching in a two-sided queueing market with unknown demand and supply curves, proposing a parameter-agnostic online policy that achieves a near-optimal Pareto frontier between profit regret and queue lengths. It combines a MaxWeight-like matching rule with a probabilistic two-price learning scheme, leveraging a two-point zero-order method and a bounded-bisection pricing loop to learn prices while controlling delays. The main theoretical contribution is a regret, AvgQLen, and MaxQLen bound for , with a near-matching lower bound in a single-link case, establishing near-optimality within a broad policy class. The numerical results corroborate the theoretical trade-offs and show substantial queue-length improvements over prior learning-based approaches, with scalable performance across multi-link settings.

Abstract

We study a two-sided market, wherein, price-sensitive heterogeneous customers and servers arrive and join their respective queues. A compatible customer-server pair can then be matched by the platform, at which point, they leave the system. Our objective is to design pricing and matching algorithms that maximize the platform's profit, while maintaining reasonable queue lengths. As the demand and supply curves governing the price-dependent arrival rates may not be known in practice, we design a novel online-learning-based pricing policy and establish its near-optimality. In particular, we prove a tradeoff among three performance metrics: regret, average queue length, and maximum queue length for , significantly improving over existing results [1]. Moreover, barring the permissible range of , we show that this trade-off between regret and average queue length is optimal up to logarithmic factors under a class of policies, matching the optimal one as in [2] which assumes the demand and supply curves to be known. Our proposed policy has two noteworthy features: a dynamic component that optimizes the tradeoff between low regret and small queue lengths; and a probabilistic component that resolves the tension between obtaining useful samples for fast learning and maintaining small queue lengths.
Paper Structure (44 sections, 10 theorems, 216 equations, 12 figures, 3 algorithms)

This paper contains 44 sections, 10 theorems, 216 equations, 12 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Main Results
  5. Algorithm
  6. Two-Point Zero-Order Method
  7. Bisection Search Method
  8. Probabilistic Two-Price Policy
  9. Computational Complexity
  10. Numerical Results
  11. An Estimate-Then-Optimize Algorithm
  12. Formal Statements of Theoretical Results
  13. Assumptions
  14. Theoretical Bounds
  15. Discussion
  16. ...and 29 more sections

Key Result

Lemma 1

Let Assumption assum:1 and Assumption assum:2 hold. Assume that the functions $-xF(x)$ (negative revenue) and $xG(x)$ (cost) are both strongly convex. Also assume the unique optimal solution to the fluid optimization problem $\lambda^*=\mu^* \notin \{0,1\}$ to avoid the trivial case. Fix a $\gamma \

Figures (12)

  • Figure 1: The model, an example with 3 types of customers and 2 types of servers.
  • Figure 2: Comparison among the two-price policy (no learning, known demand and supply functions) varma2023dynamic, the threshold policyyang2024learning, and the proposed probabilistic two-price policy with $w=0.001$. The shaded area is $95\%$ confidence interval with 10 independent runs.
  • Figure 3: Comparison among the two-price policy (no learning, known demand and supply functions) varma2023dynamic, the threshold policyyang2024learning, and the proposed probabilistic two-price policy with $w=0.01$, in a single-link system.
  • Figure 4: Comparison of regret, average queue length, and maximum queue length among the two-price policy (no learning, known demand and supply functions) varma2023dynamic, the threshold policyyang2024learning, and the proposed probabilistic two-price policy, in a single-link system.
  • Figure 5: Tradeoff between regret and average queue length.
  • ...and 7 more figures

Theorems & Definitions (10)

  • Lemma 1
  • Theorem 1
  • Corollary 1
  • Lemma 2: yang2024learning
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6: yang2024learning
  • Lemma 7: yang2024learning
  • Lemma 8: yang2024learning