Title: Transfer Faster, Price Smarter: Minimax Dynamic Pricing under Cross-Market Preference Shift
Venue: NeurIPS 2025
Accepted: yes
Keywords: Dynamic Pricing

Transfer Learning

Random Utility Models

Posterior Shift

RKHS
Abstract: We study contextual dynamic pricing when a target market can leverage $K$ auxiliary markets—offline logs or concurrent streams—whose *mean utilities differ by a structured preference shift*. We propose *Cross-Market Transfer Dynamic Pricing (CM-TDP)*, the first algorithm that *provably* handles such model-shift transfer and delivers minimax-optimal regret for *both* linear and non-parametric utility models.

For linear utilities of dimension $d$, where the *difference* between source- and target-task coefficients is $s_{0}$-sparse, CM-TDP attains regret $\tilde{\mathcal{O}}\bigl((dK^{-1}+s_{0})\log T\bigr)$. For nonlinear demand residing in a reproducing kernel Hilbert space with effective dimension $\alpha$, complexity $\beta$ and task-similarity parameter $H$, the regret becomes $\tilde{\mathcal{O}}\bigl(K^{-2\alpha\beta/(2\alpha\beta+1)}T^{1/(2\alpha\beta+1)} + H^{2/(2\alpha+1)}T^{1/(2\alpha+1)}\bigr)$, matching information-theoretic lower bounds up to logarithmic factors. The RKHS bound is the first of its kind for transfer pricing and is of independent interest.

Extensive simulations show up to 38\% higher cumulative revenue and $6\times$ faster convergence relative to single-market pricing baselines. By bridging transfer learning, robust aggregation, and revenue optimization, CM-TDP moves toward pricing systems that *transfer faster, price smarter*.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies a dynamic pricing problem with additional data from auxiliary markets, where each auxiliary market has mean utilities differing by a structured preference shift. To address this, the paper introduces the Cross-Market Transfer Dynamic Pricing (CM-TDP) algorithm, which is designed to accommodate such model shifts. The paper provides minimax-optimal regret bounds for CM-TDP and supports the proposed method through numerical experiments demonstrating its practical effectiveness.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: The paper is generally well-written and organized. It is interesting to consider and solve a transfer learning situation for dynamic pricing problems. The proposed algorithm is supported by solid theoretical results.\n\nWeaknesses:\n1. The underlying components of CM-TDP are known techniques. My opinion is that the novelty of the algorithm lies in the debiasing part to leverage data from auxiliary markets (Algorithm 2). However, the paper lacks sufficient explanation or intuition behind the algorithm's design.\n2. One (potentially) main contribution of the paper is the solid regret analysis. However, since many components in the algorithm (e.g., MLE, LASSO-like debiasing, epoch-based learning) are well-studied techniques, it would be helpful if the paper could highlight its technical novelty in the proof compared to previous works.\n3. Some assumptions are limited. For example, it seems the algorithm lacks a mechanism to \"avoid\" different markets (where $s_0$ is large in Assumption 4) and would be fooled by such a market.  Further, assuming the knowledge of distribution function $F$ is also not  practical and there have been several works to relax that assumption (e.g., [1,2]).\n\n[1] Jianqing Fan, Yongyi Guo, and Mengxin Yu. Policy optimization using semiparametric models for dynamic pricing. Journal of the American Statistical Association, 119(545):552–564, 2024.\n\n[2] Virag Shah, Ramesh Johari, and Jose Blanchet. Semi-parametric dynamic contextual pricing. In Proc. Neural Information Processing Systems, 2019.\n\nQUESTIONS:\n1. In linear bandits, it is common to include some form of regularization when performing maximum likelihood estimation (MLE). I was wondering why Equation (6) in Algorithm 2 does not include any regularization term?\n2. It would be very helpful if the authors could clearly highlight the technical novelty in their theoretical analysis, particularly in comparison to prior work. For example, does the proof involve any new concentration inequalities that are specifically tailored to the debiasing step or the transfer learning setting?\n\nLIMITATIONS:\nSee weaknesses.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-13"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the cross-market transfer dynamic pricing problem. The proposed framework can handle both offline-to-online and online-to-online settings and can accommodate linear and nonlinear utilities through reproducing kernel Hilbert Spaces (RKHS). Furthermore, the method can work for multiple heterogeneous source markets whose mean utilities differ from the target under certain assumptions. Numerical results demonstrate the potential gain by leveraging information from multiple source markets.\n\nSTRENGTHS AND WEAKNESSES:\nThe proposed transfer learning framework under utility shifts is quite broad and can work for both online-to-online and offline-to-online regimes. Furthermore, the authors consider both the linear utility class and a general nonparametric utility class through RKHS. This yields great flexibility for the proposed framework. \n\nThe proposed work assumes the utilities differ from the target by a structured shift. If true, as shown in the numerical results, the improvement can be substantial. However, it is not clear how sensitive the proposed method is to the violation of the assumptions. Furthermore, more discussions and comparisons between the linear and nonparametric cases are needed. In general, one would expect better bounds for the linear case than the nonlinear one due to the smaller class of functions for the linear case.\n\nQUESTIONS:\n1.\tThe proposed method is for the utilities with a structured shift from the target utility. What happens if the assumption is wrong? How do we verify the assumption in practice for a safe and reliable application of the proposed method?\n2.\tFor RKHS, when the linear kernel (inner product) is used, the resulting function becomes linear. Please elaborate the results for the RKHS with the linear kernel and compare it with the results for the linear case. \n3.\tSince it doesn’t appear in the bound, how does the dimension d affect the bound for the RKHS case?  \n4.\tFor the numerical results, the highest dimension considered is 20 which is quite low. How high dimensions can the method handle? It will be helpful to include some numerical results for higher dimensions such as in hundreds or thousands. \n5.\tThe patterns for the linear and RKHS in Figures 2 and 3 are almost identical. This is somewhat surprising. More elaborations will be helpful.\n6.\tThe simulation only considered identical and sparse settings. It will be important to explore denser differences and show how stable the method is as the differences vary from sparse to dense. \n7.\tIt is important to include some results on computational complexity and time comparisons.\n\nLIMITATIONS:\nThe authors mentioned very briefly towards the end that “Future directions include automated market-similarity detection and extensions to multi-product settings”. In general, one challenging aspect is to verify that the markets follow the assumptions imposed for the proposed method. If not being able to gain from source markets, one needs to ensure at least not hurt the results without using transfer learning. More discussions on these are needed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors develop contextual dynamic pricing algorithms that can leverage auxiliary markets' data to improve the performance on the target market. The difference and similarities between different markets are illustrated by the utility functions. There are multiple algorithms developed, tackling several settings including different utility function class (linear parametric models, RKHS nonparametric ones), and different auxiliary markets' data (offline static, online concurrent).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. Very practical to transfer knowledge and purchase data from markets to markets, especially useful when the target market is small.\n2. Lie in the intersection of several frontier and trending AI/ML research areas of transfer learning, contextual dynamic pricing, multi-task reinforcement learning.\n3. The established algorithms cover multiple scenarios, including different utility function class (linear parametric models, RKHS nonparametric ones), and different auxiliary markets' data (offline static, online concurrent). \n4. The interplay between the number of auxiliary markets and similarity measure between the markets is well displayed in the derived regret bounds, providing essential insights. \n5. The theoretical results align well with the numerical ones, thus it is likely that they provide operationally meaningful insights. \n\nWeaknesses:\n1. The minimax optimality, which is important and is claimed for many times across the title, abstract, main paper and conclusions, is not supported by explicit lower bounds. Moreover, such matching lower bounds may not be straightforward to derive in the considered settings. See Question 1 for more details.\n2. Less practical and a bit restrictive as the covariate distributions and market noise distributions are the same across the target market and auxiliary markets. See Question 2 for more details. \n3. Missing technical definitions, e.g., the kernel $k$ in the input of Algorithm 3, $g^{(ag)}$, $\\Sigma$ and $N(\\lambda)$ in the statement of Theorem 8. See Questions 6 and 8 for more details.\n\nQUESTIONS:\n1. Justification of \"minimax optimality\". \nIn lines 10 - 11 (abstract), 52, 310, the authors claim \"matching information-theoretic lower bounds up to logarithmic factors\", \"matching known lower bounds\", \"matching information-theoretic lower bounds\". However, I did not find any specific lower bounds proved and provided by the authors, or referred to in the existing literature, that could support the minimax optimality of the authors' derived regret upper bounds. Note that the authors' derived upper bounds are complex in the sense that they contain multiple terms and are related to similarity parameters, e.g., $s, \\alpha, \\beta$. Therefore, I doubt whether complete (not just for special degenerated cases) matching lower bounds, as claimed by the authors, exist in the literature, or can be easily derived, for their considered settings. Could authors provide explicit lower bounds to better support their claimed minimax optimality?\n\n2. Cross-market, but with the same covariate distribution and market noise distribution. \nThe authors consider transfer dynamic pricing under different markets with differences and similarities in their utility functions. However, the covariate distributions $\\mathcal{P}_{x}$ (line 124) and the market noise distribution $F$ (line 103) are assumed to be the same across these markets. I feel this limits the practical impact of these considered settings, as markets that are different in their utility functions, are very likely to also enjoy different covariate distributions and market noise distributions. Could the authors elaborate on potential extensions towards different covariate distributions and market noise distributions?\n\n3. Explicit choice for $m_{0}$ in Algorithm 4, i.e., the number of initial and early episodes that leverage the offline static source log to implement the transfer learning. \nIn lines 153, 173 - 174, and Algorithm 4, the authors illustrate that the transfer learning is only implemented in early episodes. However, the number of these early episodes, which is denoted as $m_{0}$ in Algorithm 4, is not specified explicitly in the Theorem 9, Algorithm 4, and Theorem 10. Actually, the value of $m_{0}$ is quite essential and meaningful in theory, as it demonstrates a critical phase transition in optimally using the information in the offline historical data of other markets, and online stream data of the target market. Could the authors clarify on the choice of $m_{0}$?\n\n4. Task similarity in Linear Models as the sparsity constraints in the cross-market parameter differences. \nFor the linear model, the authors use an $\\ell_{0}$ sparsity constraint in the cross-market parameter differences. Such a modeling of similarities between utility functions can be justified. However, I wonder if other types of similarity measure, e.g., the $\\ell_{1}$ constraints, can be well incorporated into the framework of this work. Could the authors provide more discussions on this issue? \n\n5. How does the smoothness parameter $\\eta$ in line 236 influence the regrets?\n\n6. How the kernel $k$, as one of the input of Algorithm 3, is defined? How is $k$ related to the original kernel $K$?\n\n7. As the feature mapping $\\phi_{t} = (p_{t},x_{t})$ in Algorithm 3 consists of both $p_{t}$ and $x_{t}$, the $\\mathcal{H}_{k}$ (based on kernel k) in lines 2 -- 5 of Algorithm 3 should be different from the $\\mathcal{H}_{k}$ (based on kernel K) in line 225, which only concerns the covariate $x$. How is the $g$ in lines 2 -- 5 of Algorithm 3, as a function mapping from $\\phi = (p,x)$, different from and related to the utility functions $g^{(k)},k\\in\\{0\\}\\cup[K]$? \n\n8. Statement of Theorem 8.\nIn line 250, $g^{(ag)}$ is not defined, even across the paper including the Appendix. In line 250, $\\Sigma$ is not defined, but defined in the Appendix. In line 251, $N(\\lambda)$ is not defined, but defined in the Appendix. Could the authors clarify on these notations?\n\n9. Relationships between regrets in linear parametric case and RKHS nonparametric case. \nAs illustrated in lines 272 - 273, the regret upper bounds for RKHS nonparametric case can reduce to $\\tilde{O}(T^{1/2})$ in connection with the well-studied linear case. How is this regret bound compared to the $\\tilde{O}(\\log T)$ regret derived under the linear parametric case? Why are they different?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies contextual pricing where a target market is able to leverage $K$ auxiliary markets, whose utilities differ by a structured preference shift. Specifically, the paper adopts a stochastic utility model $v_t=g(x_t) + \\epsilon_t$ where $g(\\cdot)$ is an unknown mean utility function and $\\epsilon$ is a random noise. In this paper, the authors propose a Cross-Market Transfer Dynamic Pricing (CM-TDP) algorithm that achieves (a) $\\tilde{O}(d/K \\log T + s_0 \\log T)$ regret for linear utility with sparse differences, and (b) $\\tilde{O}(K^{-2αβ/(2αβ+1)}T^{1/(2αβ+1)} + H^{2/(2α+1)}T^{1/(2α+1)})$ regret for the RKHS utilities. They also conducted numerical simulations, where CM-TDP significantly outperforms single-market baselines, reducing cumulative regret by up to 50%.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths:\n\nTo the best of my (as well as theirs) knowledge, this is the first paper in the pricing community that studies the utility-model shifting effect. They provide rigorous analysis and deliver minimax regret rate for both parametric and non-parametric settings (which is also the first non-parametric regret bound for transfer learning in pricing). From a high-level point of view, this work unifies bias-correction techniques from multi-task bandits with pricing, comparing to traditional bandits that only focus on prediction error. Besides, this paper is overall well-written and clearly delivered.\n\n\n### Weakness:\n\nThe most significant concern raised from my perspective lies in these strong modeling assumptions. For instance, the context distribution is assumed to be identical over all markets, which is clearly over-simplifying the scenario. Also, the utility noise distribution is assumed known, and the sparsity parameter is known ahead of time. All of these assumptions listed above are restrictive (although somewhat justified in Sec 3 but not very convencing) and undermines the potential application as well as the extension of this work. I understand that this is the very first work considering this problem, but I highly encourage the authors to add more justification and discuss potential limits/extensions with respect to each assumption they have made.\n\nThere are not much beyond my main concern, as this work is overall self-consistent. Besides, the notations are slightly heavier than I expected.\n\nQUESTIONS:\nWhat is the computational complexity of CM-TDP in different settings (i.e. in Off-to-On and On-to-On settings, on the linear model and RKHS model)?\n\nLIMITATIONS:\nDiscussed with less sufficiency. See \"Weakness\" above.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-02"}