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Transfer Learning for Nonparametric Contextual Dynamic Pricing

Fan Wang, Feiyu Jiang, Zifeng Zhao, Yi Yu

TL;DR

This work tackles nonparametric contextual dynamic pricing when target data are limited by leveraging related pre-collected data from a source domain under covariate shift. It introduces the TLDP algorithm, which adaptively partitions the joint covariate-price space and uses source data to bootstrap exploration, achieving a regret that scales as $R \lesssim n_Q\left(n_Q+(\kappa n_P)^{(d+3)/(d+3+\gamma)}\right)^{-1/(d+3)}$ up to logarithmic factors. The authors prove a matching minimax lower bound, establishing the minimax optimality of TLDP and quantifying how transfer quality, through the transfer exponent $\gamma$ and exploration coefficient $\kappa$, influences performance. They validate the approach with synthetic and real data, showing TLDP's practical benefits in reducing target regret and its robustness to transfer settings. Overall, the paper provides a rigorous foundation for transfer learning in a continuous-action, nonparametric pricing context and offers actionable guidance for deploying cross-domain pricing strategies.

Abstract

Dynamic pricing strategies are crucial for firms to maximize revenue by adjusting prices based on market conditions and customer characteristics. However, designing optimal pricing strategies becomes challenging when historical data are limited, as is often the case when launching new products or entering new markets. One promising approach to overcome this limitation is to leverage information from related products or markets to inform the focal pricing decisions. In this paper, we explore transfer learning for nonparametric contextual dynamic pricing under a covariate shift model, where the marginal distributions of covariates differ between source and target domains while the reward functions remain the same. We propose a novel Transfer Learning for Dynamic Pricing (TLDP) algorithm that can effectively leverage pre-collected data from a source domain to enhance pricing decisions in the target domain. The regret upper bound of TLDP is established under a simple Lipschitz condition on the reward function. To establish the optimality of TLDP, we further derive a matching minimax lower bound, which includes the target-only scenario as a special case and is presented for the first time in the literature. Extensive numerical experiments validate our approach, demonstrating its superiority over existing methods and highlighting its practical utility in real-world applications.

Transfer Learning for Nonparametric Contextual Dynamic Pricing

TL;DR

This work tackles nonparametric contextual dynamic pricing when target data are limited by leveraging related pre-collected data from a source domain under covariate shift. It introduces the TLDP algorithm, which adaptively partitions the joint covariate-price space and uses source data to bootstrap exploration, achieving a regret that scales as up to logarithmic factors. The authors prove a matching minimax lower bound, establishing the minimax optimality of TLDP and quantifying how transfer quality, through the transfer exponent and exploration coefficient , influences performance. They validate the approach with synthetic and real data, showing TLDP's practical benefits in reducing target regret and its robustness to transfer settings. Overall, the paper provides a rigorous foundation for transfer learning in a continuous-action, nonparametric pricing context and offers actionable guidance for deploying cross-domain pricing strategies.

Abstract

Dynamic pricing strategies are crucial for firms to maximize revenue by adjusting prices based on market conditions and customer characteristics. However, designing optimal pricing strategies becomes challenging when historical data are limited, as is often the case when launching new products or entering new markets. One promising approach to overcome this limitation is to leverage information from related products or markets to inform the focal pricing decisions. In this paper, we explore transfer learning for nonparametric contextual dynamic pricing under a covariate shift model, where the marginal distributions of covariates differ between source and target domains while the reward functions remain the same. We propose a novel Transfer Learning for Dynamic Pricing (TLDP) algorithm that can effectively leverage pre-collected data from a source domain to enhance pricing decisions in the target domain. The regret upper bound of TLDP is established under a simple Lipschitz condition on the reward function. To establish the optimality of TLDP, we further derive a matching minimax lower bound, which includes the target-only scenario as a special case and is presented for the first time in the literature. Extensive numerical experiments validate our approach, demonstrating its superiority over existing methods and highlighting its practical utility in real-world applications.

Paper Structure

This paper contains 20 sections, 8 theorems, 152 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. List of contributions
  3. Notation and organization
  4. Problem formulation
  5. Transfer learning algorithm
  6. Minimax optimality
  7. Numerical experiments
  8. Simulation studies
  9. Real data analysis
  10. Conclusion
  11. Comparisons with non-transfer learning regret bounds
  12. Proof of Theorem \ref{['theorem_upper_bound']}
  13. Proof of Theorem \ref{['theorem_upper_bound']}
  14. Auxiliary results
  15. Proof of Theorem \ref{['theorem_lower_bound']}
  16. ...and 5 more sections

Key Result

Theorem 1

Suppose that the source dataset $\mathcal{D}^P = \{(X^P_t, p^P_t, Y^P_t)\}_{t=1}^{n_P}$ is defined in eq-source-data with triplets independent across time. Assume that the target dataset, defined in def-target, satisfies eq-reward, and that Assumptions ass-lipschitz and ass-target-cov hold, with $C_ and $C_r^4 c_{\gamma} c_Q \geq 8$, where $C_r > 0$ is a constant and $c_{\gamma}, c_Q >0$ are cons

Figures (4)

  • Figure 1: Results for Configuration 1 in Scenario 1. Panel (A) and (B): varying source data size $n_P$ and target data size $n_Q$, respectively. Panel (C) varying the transfer exponent $\gamma$ (top axis) and the exploration coefficient $\kappa$ (bottom axis). Panel (D): varying the index constant $C_I$ (top axis) and the exploration radius constant $C_r$ (bottom axis). For Panels (B), (C) and (D), we fix $n_Q = 10000$.
  • Figure 2: Results for Configuration 2 in Scenario 1. Panel (A) and (B): varying source data size $n_P$ and target data size $n_Q$, respectively. Panel (C) varying the transfer exponent $\gamma$ (top axis) and the exploration coefficient $\kappa$ (bottom axis). Panel (D): varying the index constant $C_I$ (top axis) and the exploration radius constant $C_r$ (bottom axis). For Panels (B), (C) and (D), we fix $n_Q = 10000$.
  • Figure 3: Results for Configuration 1 in Scenario 2. Panel (A) and (B): varying source data size $n_P$ and target data size $n_Q$, respectively. Panel (C) varying the transfer exponent $\gamma$ (top axis) and the exploration coefficient $\kappa$ (bottom axis). Panel (D): varying the index constant $C_I$ (top axis) and the exploration radius constant $C_r$ (bottom axis). For Panels (B), (C) and (D), we fix $n_Q = 10000$.
  • Figure 4: Results for Configuration 2 in Scenario 2. Panel (A) and (B): varying source data size $n_P$ and target data size $n_Q$, respectively. Panel (C) varying the transfer exponent $\gamma$ (top axis) and the exploration coefficient $\kappa$ (bottom axis). Panel (D): varying the index constant $C_I$ (top axis) and the exploration radius constant $C_r$ (bottom axis). For Panels (B), (C) and (D), we fix $n_Q = 10000$.

Theorems & Definitions (18)

  • Definition 1: Transfer exponent
  • Definition 2: Exploration coefficient
  • Example 1
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Corollary 3
  • proof
  • Lemma 4
  • proof
  • ...and 8 more