Contextual Online Pricing with (Biased) Offline Data

We thank the reviewer for recognizing our paper’s contribution and strengths, and for the constructive comments. We provide our detailed responses below.

In what follows, we use [1] [2] etc. to refer to papers cited in our submission, and [a] [b] etc. for new references with bibliographic information given at the end of this response.

Comment 1: The ideas behind the algorithm are not presented clearly and the algorithm does not explicitly define the parameter estimators or the widths of the confidence ellipsoids in the main paper.

We thank the reviewer for the suggestion. We do outline the key ideas behind Algorithms 1–3 before presenting their regret bounds in Sections 3 and 4. The intuition for the parameter estimators and confidence‑ellipsoid widths is explained in the main paper; the full mathematical details are deferred to Appendix A due to space constraints. In the revised paper, we will expand the algorithmic details and provide additional discussions to further improve the clarity.

Comment 2: The fact that $\widehat{p}$ mimics the pricing strategy $p_n$ instead of estimating the parameter $\theta'$ is not discussed.

We thank the reviewer for this question. When the offline data are used solely to estimate $\theta_*'$ and only the bias bound $V$ is known, the squared estimation error of the online parameter $\theta_*$ cannot, in order, improve beyond $V^2+ 1/\lambda_{\min}(\hat \Sigma)$, which yields the worst-case regret bound discussed in the general price-elasticity setting.

In the scalar price-elasticity case—a key contribution of our work—the regret can be further tightened through the instance-dependent parameter $\delta^{2}$, which measures how far the empirical pricing rule $\hat p$ departs from the (unknown) optimal policy. If we understand correctly, the reviewer’s $p_n$ refers to the offline prices $\hat p_n$. Our $\hat p$ mimics these prices in a structured way: it is the best-fit linear rule $\hat p(x,y)=\hat A^\top x/y$ over the offline sample, and, when no context is present, reduces to their empirical mean. Crucially, $\hat{p}$ is not obtained by first estimating $\theta_*'$ and then selecting the corresponding optimal pricing strategy.

Then, The effect of $\delta^{2}$ (see Lines 184–206) is two-fold:

1. If $\delta^{2}$ is large, allowing online price strategies $p_t$ to diverge from $\widehat{p}$ accelerates both exploration and exploitation.
2. If $\delta^{2}$ is small, the online phase should simply adopt $\widehat{p}$.

Thus, $\widehat{p}$ guides the online strategy either to follow it or to deviate from it. Because $\delta^{2}$ is unknown, Algorithm 1 first estimates it (Line 1) and then decides whether it is small enough to justify using $\widehat{p}$ exclusively in the online phase. We will add further intuition about the structure of $\hat{p}$ in the revised paper.

Comment 3: There may be a mistake in the proof/statement of the lower bound (Theorem 2)... This would imply a total regret that is at most (poly-)logarithmic in $T$. This error was already present in [6] and [28].

We thank the reviewer for this question. We want to point out that our proof and statement of Theorem 2 is correct. First, we would like to emphasize that dynamic pricing with a linear demand model is not a purely exploitative task. Prior work [a] shows that even in the simplest non‑contextual, one‑dimensional setting, $D = \alpha + \beta \cdot p + \varepsilon,$ the regret is lower‑bounded by $\Omega(\sqrt{T})$.

The key insight is that prices do not provide the same amount of information. In dynamic pricing, the instantaneous regret satisfies $r_t \lesssim (p_t - p^*)^{2} \overset{(1)}{\lesssim} ||\theta_{\*}- \hat{\theta}_t ||^2.$

On one hand, if $\lVert\theta_*-\hat\theta_t\rVert = O(t^{-1/2})$, achieving an instantaneous regret of $O(t^{-1})$ requires that inequality (1) holds—i.e., $p_t$ must be the optimal price with respect to $\hat\theta_t$, which implies no exploration should be incorporated. On the other hand, achieving the estimation rate $\lVert\hat\theta_t-\theta_*\rVert = O(t^{-1/2})$ demands at least $\Theta(t)$ exploratory rounds, and effective exploration must use at least two prices separated by a fixed gap, incurring $\Theta(t)$ regret (as in the ILS‑d algorithm [a]). Hence, one cannot simultaneously maintain a $t^{-1/2}$ estimation error and a $t^{-1}$ instantaneous regret for all $1 \leq t \leq T$. Prior work on the estimation–regret trade‑off—see, e.g., [b]—likewise proves that no algorithm can achieve both the optimal $\sqrt{T}$ regret and the optimal $1/\sqrt{T}$ estimation error within $T$ rounds, highlighting the fundamental exploration–exploitation tension in dynamic pricing with a linear demand model.

Comment 4: The presentation of the theoretical results would benefit from greater clarity; key steps in the proofs are difficult to follow and lack structure.

We thank the reviewer for the suggestion. In the revised paper, we will expand the discussion accompanying each theoretical result and restructure the proofs.

Comment 5: The paper tackles a pricing model that deviates from the classical binary‑demand framework without explaining why, raising doubts about the scope of its contributions. The literature review is sparse—relying mainly on the outdated reference [10]. The authors should clarify how their setting relates to, or generalizes, these binary‑demand framework.

We thank the reviewer for raising this point. We would like to point out that linear‑demand models are long‑standing classics [a, e, f] and remain an active research area, as studied by recent work[2, 4, b, c, d]. Prior work [e] also demonstrates that many non‑linear demand functions can be closely approximated by a linear specification. In the revision, we will expand our discussion to clarify how our framework relates to the binary‑demand literature and highlight the connections between the two approaches. For a detailed account of our contributions, please see our response to Comment 6.

Comment 6: 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.

We would like to highlight three principal originalities of this work:

1. Instance‑dependent parameter $\delta^{2}$. We introduce the first instance‑dependent quantity, $\delta^{2}$, for contextual online pricing with offline data—whether biased or unbiased—without making any assumptions about how the offline data are generated. This parameter enables us to derive the first matching instance‑dependent lower and upper regret bounds, which are substantially tighter than existing worst‑case guarantees. The definition of $\delta^{2}$ appears on page 5 (immediately before Section 3.1), and the techniques used to obtain the matching bounds are detailed in Section 3.1 and Appendix C.

2. Extension to stochastic linear bandits. Although this paper centers on contextual online pricing, the techniques and results in Sections 3–4.1 readily extend to stochastic linear bandits with adversarially chosen action sets, as noted in lines 261–263 and detailed in Appendix I.

3. Robustness without discrepancy information. To the best of our knowledge, this is the first work to establish theoretical robustness guarantees for online learning with biased offline data when no information is available about the discrepancy between the online environment and the offline sample (see Section 4.2). Although we focus on contextual pricing, our analysis provides a clear blueprint for extending these results to a wider range of online‑learning problems.

While we draw inspiration from earlier studies on contextual online pricing with unbiased offline data [6, 28] and K‑armed bandits with biased offline data [9], the proofs required to achieve the three principal originalities above demand several novel techniques.

References:

[a] Keskin, N Bora and Zeevi, Assaf. Dynamic pricing with an unknown demand model: Asymptotically optimal semi-myopic policies. 2014.

[b] Simchi-Levi, David and Wang, Chonghuan. Pricing experimental design: causal effect, expected revenue and tail risk. 2023.

[c] Li, Xiaocheng and Zheng, Zeyu. Dynamic pricing with external information and inventory constraint. 2024.

[d] Ao, Ruicheng and Jiang, Jiashuo and Simchi-Levi, David. Learning to Price with Resource Constraints: From Full Information to Machine-Learned Prices. 2025

[e] Besbes, Omar and Zeevi, Assaf. On the (surprising) sufficiency of linear models for dynamic pricing with demand learning. 2015.

[f] Qiang, Sheng and Bayati, Mohsen. Dynamic pricing with demand covariates. 2016

I remain unconvinced by the authors’ response to Comment 3 and will maintain my score unless they provide a substantially more compelling rebuttal.

Without previous observations, the estimation error $\Vert \theta - \hat{\theta} \Vert$ decreases as (up to log factors and multiplicative constants) the square root of the inverse of the minimum eigenvalue of the design matrix $\Sigma_t$ (see, e.g., the proof of Lemma 5). Here, we have \Sigma_t = \sum_{s\leq t} z_s z_s^{\top} = \sum_{s\leq t} z'_s z'_s^{\top} + \sum_{s\leq t} z''_s z''_s^{\top} \geq \sum_{s\leq t} z'_s z'_s^{\top} where $z_s = (x_s^{\top}, p_s y_s^{\top})^{\top}$, $z'_s = (x_s^{\top}, l y_s^{\top})^{\top}$, and $z''_s = (0, (p_s - l) y_s^{\top})^{\top}$ (here the inequality should be understood in the sense $A \geq B$ if $A - B$ is semi-definite non-negative, and it holds because $p_s \geq l$). Thus, the estimation error $\Vert \theta - \hat{\theta}_t \Vert$ decreases faster than (up to log factors and multiplicative constants) the square root of the inverse of the minimum eigenvalue of the design matrix \geq \sum_{s\leq t} z'_s z'_s^{\top}, thus decreasing faster than (up to log factors and numerical constants) $1/\sqrt{t}$ (because of the assumption that $x_t$ and $y_t$ are i.i.d. with positive covariance matrices).

Thus, contrary to the author's claim, no matter the strategy (as long as it posts price above $l$), the estimation error $\Vert \theta - \hat{\theta}_t \Vert$ is of order $1/\sqrt{t}$ (here again forgetting multiplicative constants and log factors).

I strongly recommend that the authors address this point using rigorous mathematical proofs rather than informal intuition—for instance, by explicitly explaining the logical connection between the two equations immediately following Step 1.1. I am aware that this observation implies the lower bound in [b] is incorrect.

We thank the reviewer for recognizing our paper’s contribution and strengths, and for the constructive comments. We provide our detailed responses below.

In what follows, we use [1] [2] etc. to refer to papers cited in our submission, and [a] [b] etc. for new references with bibliographic information given at the end of this response.

Comment 1: Add discussions and explanations on when V can be known in practice

We thank the reviewer for raising this point. We emphasize that $V$ is only a valid upper bound on the bias $\lVert\theta_*-\theta_*'\rVert$; specifically, $V \ge \lVert\theta_*-\theta_*'\rVert$. It is not intended as an exact estimate, i.e., we do not assume $V = \lVert\theta_*-\theta_*'\rVert$. In practice, $V$ can be estimated with robust machine‑learning methods [5] or through cross‑validation [8], as also argued in prior work on K‑armed bandits with biased offline data [9]. Setting $V=\infty$ trivially guarantees validity and recovers the same performance as classical algorithms that ignore offline data (see Sections 3–4.1). However, when a finite upper bound that scales favourably with the time horizon $T$ is available, the regret guarantees in Sections 3–4.1 become sharper. We will expand the discussion that estimating a valid upper bound $V$ is practically feasible.

Comment 2: In 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?

We thank the reviewer for the question. As explained in Comment 1, our results in Sections 3–4.1 require only that $V$ be a valid upper bound on the true bias $\lVert\theta_*-\theta_*'\rVert$; such a upper bound can be estimated with robust ML methods or cross‑validation. Theorems 1 and 3 show that regret depends on $V$ and the dependence is solely through how $V$ scales with the time horizon $T$. For example, if $\lVert\theta_*-\theta_*'\rVert = T^{-5/16}$ and we use the upper bound $V = 1.1\lVert\theta_*-\theta_*'\rVert>\lVert\theta_*-\theta_*'\rVert$, then $V$ still scales as $T^{-5/16}$, and our theorems guarantee regret is strictly better than classical algorithms that ignore offline data. Numerical experiments corroborate this: with a tight estimate ($V = 1.1\lVert\theta_*-\theta_*'\rVert$) we achieve lower regret; and even with a loose estimate ($V = 10\lVert\theta_*-\theta_*'\rVert$) the regret does not exceed that of a purely online baseline.

We acknowledge that robust ML methods or cross‑validation can, with small probability, produce an invalid estimate of $V$ (i.e., $V < \lVert\theta_*-\theta_*'\rVert$). In that event, our confidence intervals no longer hold and the regret can become linear in theory. From a practical standpoint, however, it is still valuable to understand how regret behaves when $V$ is underestimated. Accordingly, our revised paper will include additional experiments that test the algorithm’s performance under such misspecifications.

Comment 3: Although 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.

We thank the reviewer for raising this point. Our linear‑model framework extends naturally to a generalized linear model with a known link function $g$: $D=g\bigl(\alpha^{\top}x + (\beta^{\top}y)p\bigr) + \varepsilon,$ provided that (i) $g$ is differentiable and strictly increasing, and (ii) the minimizer of $\bar D(p)=g\bigl(\alpha^{\top}x+ (\beta^{\top}y)p\bigr)$ lies in the interior of the feasible price interval $[l,u]$. These assumptions are standard; see [2,b]. The only change required is to replace the least‑squares estimate with a maximum quasi‑likelihood objective. Because $g$ is differentiable and strictly increasing, it is locally linear, we are confident that our core regret analysis framework can be extended to this setting and achieve comparable results; see [c] for a similar argument and [2] for the full derivation. We will add the discussion in the revised manuscript.

Comment 4: For 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.

We appreciate the reviewer’s observation. We do not claim that RCO3 always outperforms UCB. In fact, our impossibility result shows that, without any knowledge of the mismatch between offline and online data distributions, no algorithm can always outperform a purely online baseline such as UCB.

The robustness section therefore has a different goal: it introduces an algorithm that, when an upper bias bound $V$ is unavailable, (i) achieves lower regret than a pure‑online method when the true bias is small, and (ii) still guarantees sublinear regret when the bias is large. Empirically, Figure 2(c) shows that the percentage drop in regret for small bias is comparable to the percentage increase for large bias, even though the absolute loss in the latter case is higher. Theoretically, these robustness results complement Sections 3–4.1—which assume $V$ is known—by providing the first robustness guarantee for contextual online pricing with biased offline data and paving the way for analogous guarantees in broader online‑learning settings that leverage such data.

Comment 5: The 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.

We appreciate the reviewer’s suggestion. For additional discussion on how a valid upper bound $V$ can be obtained, please see our responses to Comments 1 and 2.

Usability of offline data

• If a valid bound $V$ is available: We should always incorporate the offline data and use Algorithms 1 or 2. As indicated by Theorems 1 and 3, these algorithms never perform worse than a pure‑online baseline and can outperform it when $V$ is small and the offline data are informative (Sections 3–4.1).

• If no valid bound $V$ is available: We can adopt Algorithm 3, selecting the length of the testing phase within the choice set in Theorem 5, according to the decision maker’s risk tolerance. A shorter test phase lowers regret when the true bias is small but increases the worst‑case regret, and vice versa. A special case is $\alpha = 1/2$, where the decision maker does not accept regret exceeding $\sqrt{T}$; in this scenario, Algorithm 3 matches the performance of a purely online algorithm.

We will clarify these guidelines and expand the discussion in the revised manuscript.

For discussions on potential negative societal impact, our work is primarily theoretical and focuses on the design and analysis of an online learning algorithm for pricing with biased offline data, with the goal of achieving low regret. As such, we abstract away from specific application domains, and do not make assumptions about user demographics or other sensitive attributes. We also want to emphasize that our algorithm does not rely on access to sensitive features.

That said, we agree that pricing mechanisms, even when designed with performance guarantees, can have real-world implications on fairness and equity. In future work, it would be valuable to study how such algorithms perform in settings where users belong to different groups, and to examine whether the algorithm's decisions introduce or exacerbate disparities across these groups. We will add a discussion in the revision to acknowledge these concerns and point out directions for fairness-aware extensions of our work.

Comment 6: The paper does not study real applications.

We thank the reviewer for raising this point. We agree that incorporating real-world data is an important next step. The CPRM-12-001: On-Line Auto Lending dataset from Columbia University’s Center for Pricing and Revenue Management—widely used in personalized lending-rate studies [2, d]—is a natural choice for our online component. We will perform the experiment with this (online) dataset and synthetic (offline) data generated by large language models (LLMs), which can closely approximate real-world behaviors.

We also would like to emphasize that the primary contribution of this paper is a comprehensive theoretical understanding of contextual online pricing with biased offline data. For a detailed summary of these theoretical contributions and their originality, please refer to our response to Reviewer 4jmU’s Comment 2.

References:

[a] Lei, Yanzhe and Miao, Sentao and Momot, Ruslan. Privacy-preserving personalized revenue management. 2024.

[b] Den Boer, Arnoud V and Zwart, Bert. Simultaneously learning and optimizing using controlled variance pricing. 2014.

[c] Keskin, N Bora and Zeevi, Assaf. Dynamic pricing with an unknown demand model: Asymptotically optimal semi-myopic policies. 2014.

We appreciate the reviewer’s follow-up comments and agree that broadening our discussion of nonlinear demand models will strengthen the paper. Notably, prior work [a] shows that—under a suitable regularity condition on the unknown nonlinear link function (Assumption 1 in [a])—a semi-myopic policy based on a linear model converges to the optimal policy for the true nonlinear model and achieves minimax-optimal regret. This finding further underscores the importance of studying linear models in online pricing. We will incorporate these insights and their implications into the revised manuscript.

References:

[a] Besbes, Omar and Zeevi, Assaf. On the (surprising) sufficiency of linear models for dynamic pricing with demand learning. 2015.

We thank the reviewer for recognizing our paper’s contribution and strengths, and for the constructive comments. We provide our detailed responses below.

Comment 1: 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.

We thank the reviewer for this helpful suggestion. Assumption 1 collects the standard conditions used in contextual online pricing (see, e.g., [2, 4, 16]). Because of space constraints, we grouped them into a single block. In the revised paper, we will expand the accompanying discussion to provide additional context for Assumption 1.

Comment 2: The originality of the work is limited.

We would like to highlight three main originalities of this work:

1. Instance‑dependent parameter $\delta^{2}$. We introduce the first instance‑dependent quantity, $\delta^{2}$, for contextual online pricing with offline data—whether biased or unbiased—without making any assumptions about how the offline data are generated. This parameter enables us to derive the first matching instance‑dependent lower and upper regret bounds, which are substantially tighter than existing worst‑case guarantees. The definition of $\delta^{2}$ appears on page 5 (immediately before Section 3.1), and the techniques used to obtain the matching bounds are detailed in Section 3.1 and Appendix C.

2. Extension to stochastic linear bandits. Although this paper centers on contextual online pricing, the techniques and results in Sections 3–4.1 readily extend to stochastic linear bandits with adversarially chosen action sets, as noted in lines 261–263 and detailed in Appendix I.

3. Robustness without discrepancy information. To the best of our knowledge, this is the first work to establish theoretical robustness guarantees for online learning with biased offline data when no information is available about the discrepancy between the online environment and the offline sample (see Section 4.2). Although we focus on contextual pricing, our analysis provides a clear blueprint for extending these results to a wider range of online‑learning problems.

While we draw inspiration from earlier studies on contextual online pricing with unbiased offline data [6, 28] and K‑armed bandits with biased offline data [9], the proofs required to achieve the three main originalities above demand several novel techniques.

Comment 3: 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.

We appreciate the reviewer’s recognition of our theoretical contribution and agree that broader empirical evidence would further strengthen the paper. In the revision, we will extend the numerical study by (i) exploring a wider range of online‑offline scenarios and (ii) examining the sensitivity of our results to the bias bound $V$.

Comment 4: References are often imprecise. For example: [9] -> ICML, [11] -> ICLR, [12] -> ECB, [23] -> Single author, [29] -> ICML.

We thank the reviewer for raising this point. We will correct the imprecise references in the revised manuscript.

Comment 5: 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.

We thank the reviewer for the excellent suggestion. We will provide additional details and discussions about our algorithms in the revised paper.

We thank the reviewer for recognizing our paper’s contribution and strengths, and for the constructive comments. We provide our detailed responses below.

In what follows, we use [1] [2] etc. to refer to papers cited in our submission, and [a] [b] etc. for new references with bibliographic information given at the end of this response.

Comment 1: The paper does not discuss computational aspects of the proposed algorithms.

We appreciate the reviewer’s question. The computational complexity of our algorithms indeed merits additional discussion. In Algorithms 1 and 2, we update the policy and parameter estimates by repeatedly solving a quadratically constrained quadratic program (QCQP) (see Line 7 of Algorithm 1). Although efficient QCQP solvers exist [g], we still incur $T$ oracle calls over a horizon of length $T$.

As noted by [f], fully adaptive, per-round updates hinder parallelism and limit large-scale deployment. In purely online settings, recent work on $K$-armed bandits [a], linear bandits [f], and generalized linear bandits [e] shows that optimal $O(\sqrt{T})$ regret can be achieved with only $O(\log\log T)$ policy updates. Adapting these ideas to our setting—developing regret-optimal methods that require far fewer QCQP calls—is a promising research direction. We will expand on this avenue in the revised manuscript.

Comment 2: Proposed methods assume the linear demand model is correctly specified, there is no analysis or experimental evaluation of robustness to model misspecification.

We thank the reviewer for this suggestion. Developing algorithms that are robust to model misspecification—for example, those explored in [b, c]—is a popular research direction. We note, however, that even with unbiased offline data, the effects of model misspecification are not yet fully understood. A promising avenue for future work is to establish theoretical guarantees for online pricing when both unbiased offline data and an unknown (potentially misspecified) demand model are present. Investigation of this problem requires new ideas for algorithm design and theoretical analysis, which is beyond the scope of the current paper. We will leave it as future work and expand this discussion in the revised manuscript.

Comment 3: 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. The authors do not explicitly discuss limitations related to real-world deployment.

We thank the reviewer for raising this point. We agree that incorporating real-world data is an important next step. The CPRM-12-001: On-Line Auto Lending dataset from Columbia University’s Center for Pricing and Revenue Management—widely used in personalized lending-rate studies [2, d]—is a natural choice for our online component. We will perform the experiment with this (online) dataset and synthetic (offline) data generated by large language models (LLMs), which can closely approximate real-world behaviors.

We also wish to emphasize that the primary contribution of this paper is a comprehensive theoretical understanding of contextual online pricing with biased offline data. The assumptions on context distributions, noise, and demand patterns are standard in the literature [2] for deriving theoretical results of online pricing, and our simulations serve to validate the theory.

Comment 4: Could the authors provide guidance on choosing the length of the test phase in RCO3 and its sensitivity to this choice?

We thank the reviewer for the question. As Theorem 5 and its discussion show, the test‑phase length is not uniquely optimal; any value within a certain range is admissible. Within this range, a shorter test phase lowers regret when the true bias is small but increases the worst‑case regret. Therefore, the choice ultimately depends on the decision maker’s risk tolerance—balancing a higher worst‑case regret against the potential for better performance when the bias is small. A special case is $\alpha = 1/2$, where the decision maker does not accept regret exceeding $\sqrt{T}$; in this scenario, Algorithm 3 matches the performance of a purely online algorithm. We will expand the manuscript to clarify this trade‑off.

Comment 5: 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?

We thank the reviewer for the question. In the scalar price‑elasticity case we assume i.i.d. contexts, which allows us to define the instance‑dependent parameter $\delta^{2}$ and derive the corresponding instance‑dependent regret bound. For worst‑case regret bounds, however, this assumption is unnecessary; our results still hold when contexts are chosen adversarially. Lines 261–263 and Appendix I already show how our techniques extend to stochastic linear bandits with adversarially selected action sets. We will clarify the role of the i.i.d. assumption in the revised manuscript.

Comment 6: The 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.

We thank the reviewer for raising this important point. Our work is primarily theoretical and focuses on the design and analysis of an online learning algorithm for pricing with biased offline data, with the goal of achieving low regret. As such, we abstract away from specific application domains, and do not make assumptions about user demographics or other sensitive attributes. We also want to emphasize that our algorithm does not rely on access to sensitive features.

That said, we agree that pricing mechanisms, even when designed with performance guarantees, can have real-world implications on fairness and equity. In future work, it would be valuable to study how such algorithms perform in settings where users belong to different groups, and to examine whether the algorithm's decisions introduce or exacerbate disparities across these groups. We will add a discussion in the revision to acknowledge these concerns and point out directions for fairness-aware extensions of our work.

References:

[a] Perchet, Vianney and Rigollet, Philippe and Chassang, Sylvain and Snowberg, Erik. Batched bandit problems. 2016

[b] Besbes, Omar and Zeevi, Assaf. On the (surprising) sufficiency of linear models for dynamic pricing with demand learning. 2015.

[c] Nambiar, Mila and Simchi-Levi, David and Wang, He. Dynamic learning and pricing with model misspecification. 2019.

[d] Lei, Yanzhe and Miao, Sentao and Momot, Ruslan. Privacy-preserving personalized revenue management. 2024.

[e] Sawarni, Ayush and Das, Nirjhar and Barman, Siddharth and Sinha, Gaurav. Generalized linear bandits with limited adaptivity. 2024.

[f] Ruan, Yufei and Yang, Jiaqi and Zhou, Yuan. Linear bandits with limited adaptivity and learning distributional optimal design. 2021

[g] Nesterov, Yurii and Nemirovskii, Arkadii. Interior-point polynomial algorithms in convex programming. 1994.

We appreciate the reviewer’s follow-up comments. We would like to highlight four main originalities of this work:

1. Instance‑dependent parameter $\delta^2$. We introduce the first instance‑dependent quantity $\delta^2$, for contextual online pricing with offline data—whether biased or unbiased—without making any assumptions about how the offline data are generated. This parameter enables us to derive the first matching instance‑dependent lower and upper regret bounds, which are substantially tighter than existing worst‑case guarantees.

2. Biased offline data. We emphasize that the bias bound $V$ introduces substantial challenges in deriving tight upper and lower bounds—especially when our goal is to obtain instance-dependent results. In particular, correctly figuring out the respective roles of $\delta^{2}$ and $V$ adds an extra layer of difficulty to the analysis.

Regarding the first two points, we emphasize that the paper introduces both a novel algorithmic design and new proof techniques.

• Algorithmic design. We develop a three-ellipsoid confidence-interval construction, coupled with a carefully chosen test period, to obtain a tight upper bound.
• Proof techniques. The above two innovations create substantial challenges in deriving matching upper and lower bounds, far beyond those encountered in the analyses of Bu et al. (2020) and [28].

We would also like to emphasize some completely novel results of this paper:

3. Extension to stochastic linear bandits. Although this paper centers on contextual online pricing, the techniques and results in Sections 3–4.1 readily extend to stochastic linear bandits with adversarially chosen action sets, as noted in lines 261–263 and detailed in Appendix I.

4. Robustness without discrepancy information. To the best of our knowledge, this is the first work to establish theoretical robustness guarantees for online learning with biased offline data when no information is available about the discrepancy between the online environment and the offline sample (see Section 4.2). Although we focus on contextual pricing, our analysis provides a clear blueprint for extending these results to a wider range of online‑learning problems.

We will state our contributions more clearly in the revised manuscript.