Title: Contextual Online Pricing with (Biased) Offline Data
Venue: NeurIPS 2025
Accepted: yes
Keywords: online pricing

offline data

distributional shift

robustness
Abstract: We study contextual online pricing with biased offline data.  For the scalar price elasticity case, we identify the instance-dependent quantity $\delta^2$ that measures how far the offline data lies from the (unknown) online optimum. We show that the time length $T$, bias bound $V$,  size $N$ and dispersion $\lambda_{\min}(\hat{\Sigma})$ of the offline data, and  $\delta^2$  jointly determine the statistical complexity.  An Optimism‑in‑the‑Face‑of‑Uncertainty (OFU) policy achieves a minimax-optimal, instance-dependent regret bound $\tilde{\mathcal{O}}\big(d\sqrt{T} \wedge (V^2T + \frac{dT }{\lambda_{\min}(\hat{\Sigma}) + (N \wedge T) \delta^2})\big)$.  For general price elasticity, we establish a worst‑case, minimax-optimal rate $\tilde{\mathcal{O}}\big(d\sqrt{T} \wedge (V^2T + \frac{dT }{\lambda_{\min}(\hat{\Sigma})})\big)$ and provide a generalized OFU algorithm that attains it.  When the bias bound $V$ is unknown, we design a robust variant that always guarantees sub‑linear regret and strictly improves on purely online methods whenever the exact bias is small.  These results deliver the first tight regret guarantees for contextual pricing in the presence of biased offline data.  Our techniques also transfer verbatim to stochastic linear bandits with biased offline data, yielding analogous bounds.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of dynamic pricing in the presence of offline data, possibly coming from a different distribution. The model is the following.\n\nThe learner has access to $N$ offline observations $(D_n, x_n, y_n, p_n)_{n\\leq N}$, where $p_n$ is the price corresponding to the $n$-th (offline) transaction, $x_n$ and $y_n$ are contexts describing this transaction, and $D_n$ is the demand level for this sale at the price $p_n$, given by\n$$D_n = x_n^{\\top}\\alpha' + y_n^{\\top}\\beta' + \\epsilon'_n,$$\nwhere $\\epsilon'_n$ is a noise term. Then, at each round $t\\leq T$, the learner observes a context $x_t, y_t$, posts a price $p_t$, and receives a reward\n$$p_tD_t = p_t(x_t^{\\top}\\alpha + y_t^{\\top}\\beta + \\epsilon_t).$$\n\nThe authors propose a mechanism to leverage the previous observations and mimic the offline strategy to maximize the expected cumulative reward. They derive upper bounds on the regret, depending on problem-dependent constants, on the distance between the offline and online parameters $\\alpha', \\beta'$ and $\\alpha, \\beta$, and on the gap between the optimal strategy and the best linear approximation to the offline strategy. They also obtain lower bounds on the regret for this problem and present simulations showcasing their method.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\nThis paper studies the well-motivated problem of leveraging prior informations in dynamic pricing problem. This is, in my opinion, a relevant problem as classical bandit technic rely on a lot of exploration, which is problematic in practise.\n\n**Weaknesses** \n\n- *Clarity*: I find the paper’s clarity to be below the standards for NeurIPS. For example, the ideas behind the algorithm are not presented clearly. Among other issues, the fact that $\\widehat{p}$ mimics the pricing strategy $p_n$ instead of estimating the parameter $\\theta'$ is not discussed. The authors present their algorithm as addressing an exploration-exploitation trade-off, when in fact the constraints that the price should be upper and lower bounded, combined with the randomness of the context, imply that all prices yield approximately the same amount of information. Thus, the problem reduces to estimating the parameters of a linear regression model from offline and online data and so all rounds correspond to \"exploitation rounds\". Additionally, the presentation of the theoretical results would benefit from greater clarity; key steps in the proofs are difficult to follow and lack structure.  Moreover, the current version of the paper lacks rigor in certain aspects—for instance, the algorithm does not explicitly define the parameter estimators or the widths of the confidence ellipsoids. These crucial details are only provided in the proof of a lemma in the appendix, which undermines the accessibility and transparency of the method.\n\n- *Rigor*: The authors have addressed my concern regarding an error in the proof of their lower bound. \n\n- *Significance*: The authors study a problem very different from the one classically used in the literature, without justifying it, which may indicate that the contributions are limited. I strongly feel that the discussion of related literature is insufficient. From my (admittedly limited) knowledge of dynamic pricing and dynamic programming, the demand model used in this paper appears to diverge significantly from the classical framework, which typically assumes that each buyer has a private valuation for the good, and demand is binary—equal to 1 if the posted price is below the valuation, and 0 otherwise. This standard model has been extensively studied, and it would be valuable for the authors to more thoroughly engage with this line of work. In particular, it would be helpful to clarify how their setting compares with or generalizes these models. The current treatment, which mainly cites the outdated reference [10]—itself focused primarily on the deterministic case—is not sufficient in my opinion. \n\n- *Originality*: The technical improvement with regards to the previous work [6] in the one dimensional elasticity setting seems incremental at best as the authors study the same model under very similar assumptions, only assuming that the online and offline parameters may be different.In the multi-dimensional elasticity setting, the techniques appear to be standard, and the results are largely expected. Therefore, the technical contributions appear to be of limited significance.\n\nQUESTIONS:\nAdequately addressed by the authors.\n\nLIMITATIONS:\nOne of the limitations of this work is that the model considered differs significantly from the classical model of dynamic pricing with covariates, which is not properly discussed in the paper.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 1, "DATE": "2025-06-18"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors studied the problem of contextual online pricing with offline data which can be biased. Existing work showed that when offline data are unbiased in the sense of coming from the same distribution as the online data, incorporating the offline data to online policy can outperform pure online learning techniques. However, distribution shifts may exist, and it is common for the offline data to be different from the online data. As a result, it would be desirable to study the effect of potentially biased offline data for online learning. In this paper, the authors proposed an Optimism-in-the-Face-of-Uncertainty policy which is shown to be minimax-optimal. Both the scalar price elasticity and the general setting have been considered. Both known and unknown bias bounds have been investigated as well.\n\nSTRENGTHS AND WEAKNESSES:\nOnline pricing with the offline data is an interesting and worthwhile problem to study. This paper investigates the potential data shifts in terms of biased offline data compared with online data. The regret bounds shed some lights on the impact of offline data for online learning with respect to various factors such as bias bound etc. \nAlthough theoretically insightful, the majority of the paper considers the case of known V. However, in practice, V is likely unknown. Despite there are some developments for the case of unknown V, it is important and necessary for authors to add discussions and explanations on when V can be known in practice. Also, the numerical studies are somewhat limited to two simple simulated settings.\n\nQUESTIONS:\n1.\tIn the case where V is assumed to be known, how sensitive is the algorithm to inaccuracies in V? How strongly does it rely on the precise specification of V?\n2.\tFor the regret bounds for the general case, why N does not appear in the bounds?\n3.\tAlthough linear demand model was previously used in the literature and is the focus for this paper, it is unlikely to hold in practice. Some discussion on nonlinear models and the corresponding implications would be beneficial.\n4.\tFor RCO3 in the numerical study, it appears that the benefit is relatively small for small V yet the damage on regret can be very substantial for larger V. Overall, despite the authors claim robustness, it seems safer to use UCB instead.\n\nLIMITATIONS:\nThe current paper does not have clear discussions on the limitations and potential negative societal impact of their work. More discussions on the feasibility of known V and how to examine the usability of offline data are necessary. The paper does not study real applications.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies contextual online pricing where a firm must set prices over time while learning customer demand parameters from observed sales. In addition to online data, the firm has access to offline data that may be biased, meaning it comes from a different distribution than the online environment. The authors propose CO3 algorithm for scalar price elasticity, GCO3 algorithm for general elasticity, and RCO3 algorithm for settings where the bias level is unknown. They derive instance-dependent and minimax-optimal regret bounds that explicitly capture the effects of offline data bias, offline data dispersion, and online learning horizon. The paper also establishes matching lower bounds and impossibility results, showing when and how offline data can improve over purely online approaches. The theoretical results are supported by simulation experiments illustrating the performance of the proposed algorithms relative to standard baselines.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1) The paper provides a rigorous theoretical framework for contextual online pricing with biased offline data.\n\n2) The formulation of the Contextual Bandits for Online Pricing with Offline Data problem and the proposed algorithms are novel. \n\n3) The paper is clearly written and well organized.\n\n4) The paper addresses a practically important problem.\n\nWeaknesses:\n\n1) The paper does not discuss computational aspects of the proposed algorithms.\n\n2) Proposed methods assume the linear demand model is correctly specified, there is no analysis or experimental evaluation of robustness to model misspecification.\n\n3) Proposed methods are not tested on real-world datasets or realistic pricing simulations, where context distributions, noise properties, or demand patterns may not fully match the assumptions.\n\n4) The authors do not explicitly discuss limitations related to real-world deployment.\n\nQUESTIONS:\n1) Could the authors clarify the computational cost of their algorithms?\n\n2) Could the authors provide guidance on choosing the length of the test phase in RCO3 and its sensitivity to this choice?\n\n3) The paper assumes i.i.d. contexts over time. Could the authors comment on how their regret guarantees might extend or degrade under temporal dependence or adversarially chosen contexts?\n\nLIMITATIONS:\nThe paper does not address potential negative societal impacts, such as how biased offline data or aggressive pricing strategies could unintentionally amplify unfair pricing or discrimination.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 1, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe work focuses on online pricing in the presence of biased data retrieved from previous offline interactions (not made by a specific algorithm). The authors introduce the setting and the assumptions, and present 3 algorithms to learn online in this setting, exploiting also offline data. One algorithm is for scalar price elasticity, while the others are for general price elasticity. The authors study the regret guarantees of the solutions and provide an experimental validation of their solutions.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths \n- The paper is well-written and structured. It is visible that the authors invested time in the writing phase to properly present their findings in the limited space available.\n- The authors, at every step, discuss what they are presenting, comparing it with the literature.\n- The theoretical analysis is formal and well-performed, and all the results are reasonable given the reviewer's knowledge of the related literature. I haven't checked carefully the theoretical analysis, but it seems to be complete.\n\n### Weaknesses\nI haven't seen any critical points for this work. However, I have some weaknesses to highlight.\n- The authors present all the assumptions in a unique \"block\" (Assumption 1). This assumption should be better discussed to allow the user to understand it properly.\n- The originality of the work is limited.\n- The experimental validation is limited and presented in a very concentrated way. However, given the theoretical character of this work, this experimental validation is sufficient.\n\n### Minors\n\n- References are often imprecise. For example:\n    [9]  -> ICML, \n    [11] -> ICLR,\n    [12] -> ECB, \n    [23] -> Single author,\n    [29] -> ICML.\n- In order to understand the algorithms, the reader has to rely heavily on the pseudocode. The presentation of the algorithms, in my opinion, should be expanded in the camera-ready exploiting the additional page.\n\nQUESTIONS:\nSee weaknesses (and minors).\n\nLIMITATIONS:\nYes. The limitations of the work are properly discussed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}