Transfer Learning for Contextual Joint Assortment-Pricing under Cross-Market Heterogeneity

1. Problem Formulation

The paper addresses the Contextual Joint Assortment-Pricing problem in a multi-market setting under bandit feedback.

• Setting: A seller operates across multiple related markets (one target market and $H$ source markets). The goal is to maximize cumulative revenue in the target market ($h=0$) over a time horizon $T$.
• Decision Variables: At each time step, the seller must simultaneously select a subset of products (assortment $S_t \subseteq [N]$ with $|S_t| \le K$) and set continuous prices ($p_t$) for the selected items.
• Demand Model: Customer choices follow a Contextual Multinomial Logit (MNL) model. The latent utility for product $i$ in market $h$ is linear in parameters:
  $$v_{it}^{(h)} = \langle x_{it}^{(h)}, \theta^{(h)} \rangle - \langle x_{it}^{(h)}, \gamma^{(h)} \rangle p_{it}$$
  where $x_{it}$ are observed covariates, $\theta$ represents baseline preferences, and $\gamma$ represents price sensitivity.
• Feedback: The seller observes only the realized purchase (bandit feedback), not the full demand curve or latent utilities.
• The Challenge: While data from source markets can accelerate learning, cross-market heterogeneity (differences in customer preferences) can introduce systematic bias if data is pooled naively. The core question is how to safely transfer knowledge to reduce variance without incurring bias from structural differences.

2. Methodology: Transfer Joint Assortment-Pricing (TJAP)

The authors propose TJAP, a bias-aware framework built on a structured transfer learning assumption and an aggregate-then-debias strategy.

A. Structured Heterogeneity Assumption

The paper models cross-market differences not as arbitrary shifts, but as sparse utility shifts.

• Markets share a common contextual utility structure.
• Differences between the target market and any source market are confined to a sparse set of latent preference coordinates (size $s_0$).
• Mathematically, the parameter difference $\nu^{(0)} - \nu^{(h)}$ is supported on a common index set $\mathcal{S}^*$ with $|\mathcal{S}^*| \le s_0$.

B. Algorithm Components

TJAP operates in episodes of geometrically increasing length and consists of three core mechanisms:

1. Aggregate-Then-Debias Estimation:

   • Aggregation (Variance Reduction): Source market data are pooled to estimate a "shared" preference center ($\hat{\nu}^{(ag)}$). This reduces estimation variance along the shared dimensions by a factor proportional to $1+H$.
   • Debiasing (Target Adaptation): The pooled estimate is refined using only target market data via an $\ell_1$-regularized optimization. This step estimates the sparse discrepancy ($\delta^*$) between the pooled center and the true target parameter, correcting for the bias introduced by heterogeneity.
   • Result: An estimator $\hat{\nu} = \hat{\nu}^{(ag)} + \hat{\delta}$ that balances variance reduction with bias control.

2. Optimistic Decision Rule with Price-Uniform Bounds:

   • TJAP uses a UCB-style (Upper Confidence Bound) policy.
   • Two-Radius Confidence Bounds: The confidence interval for the utility function has two components:
     1. Variance Radius ($\alpha$): Shrinks with the total pooled information from all markets ($1+H$).
     2. Bias Radius ($\beta$): Accounts for the residual error from the sparse shift, scaling with the sparsity $s_0$ and target data size.
   • Price Uniformity: Since prices are continuous, the algorithm constructs confidence bounds that hold uniformly over the price domain $[0, \bar{P}]$ using a monotone-Lipschitz envelope to ensure the optimism principle holds for the revenue maximization problem.

3. Episodic Information-Geometry Control:

   • To handle the endogeneity of adaptive decisions (where choices affect future data quality), the algorithm freezes the information geometry (Fisher information matrix) within each episode.
   • It employs a forced-exploration gate: If the target market's Fisher information (curvature) is insufficient to identify the sparse shift, the algorithm triggers a short period of random exploration to ensure identifiability.

3. Key Contributions

1. Novel Framework: First framework for safe transfer learning in contextual joint assortment-pricing under MNL demand with bandit feedback.
2. Bias-Aware Estimation: Introduces an aggregate-then-debias pipeline that explicitly separates shared learning (variance reduction) from target-specific adaptation (bias control).
3. Theoretical Guarantees:
   • Establishes matching minimax regret bounds of order:
     $$\tilde{O}\left( d \sqrt{\frac{T}{1+H}} + s_0 \sqrt{T} \right)$$
   • Variance Term: $d \sqrt{T/(1+H)}$ shows that transfer accelerates learning along shared directions, with gains scaling by the number of source markets $H$.
   • Bias Term: $s_0 \sqrt{T}$ represents the irreducible cost of learning the sparse heterogeneous components, which depends only on the sparsity $s_0$ and target data, not on $H$.
4. Lower Bound: Proves a matching minimax lower bound, demonstrating that the variance-bias tradeoff is fundamental and cannot be improved upon by any policy.

4. Results and Findings

• Theoretical: The regret bounds reveal a transparent tradeoff. When heterogeneity is sparse ($s_0 \ll d$), transfer yields substantial gains. When heterogeneity is diffuse ($s_0 \approx d$), the benefits of transfer saturate, and the bias term dominates.
• Numerical Experiments:
  • Acceleration: TJAP consistently outperforms "Target-Only" learning (CAP) and "Naive Pooling" (ignoring heterogeneity).
  • Robustness: As the number of source markets ($H$) increases, regret decreases systematically, validating the $1/\sqrt{1+H}$ scaling.
  • Safety: Naive pooling (aggregating all data without debiasing) leads to significantly higher regret when markets differ, confirming the necessity of the bias-correction step.
  • Joint Optimization: TJAP outperforms pricing-only baselines, highlighting the value of jointly optimizing assortments and prices.

5. Significance

This work provides a principled solution to a critical problem in modern revenue management: how to leverage historical data from related markets without falling victim to "negative transfer" caused by structural differences.

• Operational Insight: It clarifies when transfer is beneficial (localized heterogeneity) and how to implement it safely (aggregate-then-debias).
• Theoretical Advance: It bridges the gap between high-dimensional transfer learning (statistics) and online learning with combinatorial constraints (revenue management), providing the first minimax-optimal characterization of transfer limits in joint assortment-pricing.
• Practical Impact: The framework offers a robust algorithm for digital platforms and retailers operating in multi-region or multi-segment environments, enabling faster learning and higher revenue while mitigating the risks of cross-market bias.